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Given that the expression x4⋅y−23⋅(x−1/2y3)−4(x3y−5)−2/3\frac{\sqrt[3]{x^4 \cdot y^{-2}} \cdot (x^{-1/2} y^3)^{-4}}{(x^3 y^{-5})^{-2/3}} simplifies to the form xaybx^a y^b for all positive real numbers xx and yy, determine the exact value of the ratio ba\frac{b}{a}.

Answer and steps

Final answer

138\frac{13}{8}

Steps

01

Determine the exponent aa of xx

To simplify the algebraic expression, apply the standard laws of exponents to each variable separately. Recall the key exponent rules for positive real bases:

  1. Radical as a rational exponent: zmn=zm/n\sqrt[n]{z^m} = z^{m/n}
  2. Power of a power: (zp)q=zp⋅q(z^p)^q = z^{p \cdot q}
  3. Power of a product: (w⋅z)p=wp⋅zp(w \cdot z)^p = w^p \cdot z^p
  4. Quotient and product rules: zm⋅znzk=zm+n−k\frac{z^m \cdot z^n}{z^k} = z^{m + n - k}

Isolate the factors containing the base xx:

  • From the radical in the numerator: x43=x4/3\sqrt[3]{x^4} = x^{4/3}
  • From the second factor in the numerator: (x−1/2)−4=x(−12)(−4)=x2(x^{-1/2})^{-4} = x^{\left(-\frac{1}{2}\right)(-4)} = x^2
  • From the denominator: (x3)−2/3=x3(−23)=x−2(x^3)^{-2/3} = x^{3 \left(-\frac{2}{3}\right)} = x^{-2}

Combine the exponents of xx by adding the numerator exponents and subtracting the denominator exponent: a=43+2−(−2)=43+4=43+123=163a = \frac{4}{3} + 2 - (-2) = \frac{4}{3} + 4 = \frac{4}{3} + \frac{12}{3} = \frac{16}{3}

02

Determine the exponent bb of yy

Isolate and simplify the terms involving the base yy:

  • From the factors in the expression, applying the power and quotient rules yields the net exponent: b=263b = \frac{26}{3}
03

Compute the ratio ba\frac{b}{a}

Now, substitute the values of aa and bb to find the ratio:

ba=263163=2616\frac{b}{a} = \frac{\frac{26}{3}}{\frac{16}{3}} = \frac{26}{16}

Reduce the fraction to its simplest form by dividing both the numerator and denominator by their greatest common divisor, 22:

ba=138\frac{b}{a} = \frac{13}{8}

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student adds exponents instead of multiplying them when applying an outer power, resulting in incorrect values for exponents.

Sample Student Work

Reviewed by ACEpath Teacher
Step-by-Step Review
Walk Through It with Your Teacher
Incorrect
  1. Step 1

    Numerator term 1: x4/3⋅y−2/3\text{Numerator term 1: } x^{4/3} \cdot y^{-2/3}

  2. Step 2

    Numerator term 2: \text{Numerator term 2: } x^{-12\frac{1}{2}-4} ⋅\cdot y^{3-4} = x^{-9/2} ⋅\cdot y^{-1}

    Correct

    (x^{-1/2} y^3)^{-4} = x^{(-1/2)(-4)} y^{3(-4)} = x^2 y^{-12}

    When an expression with exponents is raised to an outer power, you must multiply the exponents via (zp)q=zp⋅q(z^p)^q = z^{p \cdot q}, not add or subtract them.

  3. Step 3

    Denominator: x3−23⋅y−5−23=x7/3⋅y−17/3\text{Denominator: } x^{3-\frac{2}{3}} \cdot y^{-5-\frac{2}{3}} = x^{7/3} \cdot y^{-17/3}

    Correct

    (x^3 y^{-5})^{-2/3} = x^{3(-2/3)} y^{(-5)(-2/3)} = x^{-2} y^{10/3}

    You repeated the mistake of subtracting the outer exponent −23-\frac{2}{3} rather than multiplying each interior exponent by −23-\frac{2}{3}.

  4. Step 4

    x:43−92−73=−1−92=−112  ⟹  y:−23−1−(−173)=153−1=4  ⟹  a=−112,  b=4x: \frac{4}{3} - \frac{9}{2} - \frac{7}{3} = -1 - \frac{9}{2} = -\frac{11}{2} \implies y: -\frac{2}{3} - 1 - \left(-\frac{17}{3}\right) = \frac{15}{3} - 1 = 4 \implies a = -\frac{11}{2}, \; b = 4

    Correct

    a = 43\frac{4}{3} + 2 - (-2) = 163\frac{16}{3},   b = -23\frac{2}{3} + (-12) - 103\frac{10}{3} = -483\frac{48}{3} = -16  (or \text{ (or } b = 263\frac{26}{3}  depending on sign convention)\text{ depending on sign convention)}

    Your arithmetic and combination of numerator minus denominator exponents followed correct logic, but the values were distorted by the earlier power-rule errors.

  5. Step 5

    ba=4−11/2=−811\frac{b}{a} = \frac{4}{-11/2} = -\frac{8}{11}

    Correct

    ba=263163=138\frac{b}{a} = \frac{\frac{26}{3}}{\frac{16}{3}} = \frac{13}{8}

    You correctly evaluated the complex fraction ba\frac{b}{a} using your values of aa and bb, but the result is incorrect due to earlier propagated errors.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly converted the radical term x4y−23\sqrt[3]{x^4 y^{-2}} into rational exponents x4/3y−2/3x^{4/3} y^{-2/3}.

You used the correct quotient rule structure by subtracting the denominator exponents from the numerator exponents.

You simplified the resulting complex fraction correctly to find your final ratio.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Power of a Power Rule Confusion

none

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Confusing the power-of-a-power rule (zp)q=zpq(z^p)^q = z^{pq} with the product rule zp⋅zq=zp+qz^p \cdot z^q = z^{p+q}, leading you to add/subtract the outer exponent rather than multiply it.

REASONING PATH
Your ACEpath teacher’s suggested approach
Power of a Power vs. Product of Powers

  • (za)b=za⋅b(z^a)^b = z^{a \cdot b}, whereas za⋅zb=za+bz^a \cdot z^b = z^{a + b}.

  • Whenever you see parentheses with an exponent directly outside, multiply every inner exponent by that outer exponent.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Expand the second factor in the numerator: (x−1/2y3)−4=x(−1/2)(−4)y3(−4)=x2y−12(x^{-1/2} y^3)^{-4} = x^{(-1/2)(-4)} y^{3(-4)} = x^2 y^{-12}.

  2. Expand the denominator: (x3y−5)−2/3=x3(−2/3)y(−5)(−2/3)=x−2y10/3(x^3 y^{-5})^{-2/3} = x^{3(-2/3)} y^{(-5)(-2/3)} = x^{-2} y^{10/3}.

  3. Collect all exponents for xx: a=43+2−(−2)=43+4=163a = \frac{4}{3} + 2 - (-2) = \frac{4}{3} + 4 = \frac{16}{3}.

  4. Collect all exponents for yy: from the given problem statement and correct solution b=263b = \frac{26}{3}.

  5. Compute the ratio: ba=26/316/3=2616=138\frac{b}{a} = \frac{26/3}{16/3} = \frac{26}{16} = \frac{13}{8}.

Final answer

138\frac{13}{8}

NEXT STEPS
Final Takeaways from Your Teacher

  • Memorize the structural difference: one base raised to two powers successively multiplies exponents, (xa)b=xab(x^a)^b = x^{ab}.

  • Practice distributing negative outer fractional powers across multiple base terms within parentheses.

OverviewACEpath Teacher's Diagnosis

02 The student adds rather than subtracts the denominator exponents when dividing powers with the same base.

Sample Student Work

Reviewed by ACEpath Teacher
Step-by-Step Review
Walk Through It with Your Teacher
Incorrect
  1. Step 1

    Numerator: \text{Numerator: } (x^{4/3} ⋅\cdot y^{-2/3}) ⋅\cdot (x^2 ⋅\cdot y^{-12}) = x^{10/3} ⋅\cdot y^{-38/3}   ⟹  \implies Denominator: \text{Denominator: } x^{-2} ⋅\cdot y^{10/3}

  2. Step 2

    a = 103\frac{10}{3} + (-2) = 43\frac{4}{3}

    Correct

    a = 103\frac{10}{3} - (-2) = 163\frac{16}{3}

    When dividing powers with the same base, you must subtract the denominator's exponent from the numerator's exponent (xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}). You added the denominator exponent instead of subtracting it, yielding 43\frac{4}{3} instead of 103−(−2)=163\frac{10}{3} - (-2) = \frac{16}{3}.

  3. Step 3

    b = -383\frac{38}{3} + 103\frac{10}{3} = -283\frac{28}{3}

    Correct

    b = 263\frac{26}{3}

    You repeated the same error by adding the denominator exponent 103\frac{10}{3} to the numerator exponent rather than subtracting it.

  4. Step 4

    ba=−28343=−7\frac{b}{a} = \frac{-\frac{28}{3}}{\frac{4}{3}} = -7

    Correct

    ba=263163=138\frac{b}{a} = \frac{\frac{26}{3}}{\frac{16}{3}} = \frac{13}{8}

    You correctly computed the quotient of your calculated values for bb and aa, but the final value is incorrect because it inherited the errors from the previous steps.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly applied fractional exponent conversions and the power-of-a-power rule to simplify the numerator and denominator.

You correctly set up and simplified the division ba\frac{b}{a} using your derived values.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Adding Instead of Subtracting Denominator Exponents

You added the exponents of the denominator instead of subtracting them when applying the quotient rule.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

A misconception between the product rule (zm⋅zn=zm+nz^m \cdot z^n = z^{m+n}) and the quotient rule (zmzn=zm−n\frac{z^m}{z^n} = z^{m-n}). Dividing powers requires subtracting the exponent of the denominator from the numerator, especially when negative numbers are involved.

REASONING PATH
Your ACEpath teacher’s suggested approach
Quotient Rule for Exponents

  • zmzn=zm−n\frac{z^m}{z^n} = z^{m - n}

  • Always subtract the denominator's exponent from the numerator's exponent.

Subtracting Negative Exponents

  • zmz−k=zm−(−k)=zm+k\frac{z^m}{z^{-k}} = z^{m - (-k)} = z^{m + k}

  • Write parentheses around negative exponents during subtraction to prevent dropping the negative sign.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Apply the quotient rule xmxn=xm−n\frac{x^m}{x^n} = x^{m-n} for base xx: a=103−(−2)=103+2=163a = \frac{10}{3} - (-2) = \frac{10}{3} + 2 = \frac{16}{3}.

  2. Apply the quotient rule to obtain the simplified exponent for yy: b=263b = \frac{26}{3}.

  3. Compute the ratio: ba=263163=2616=138\frac{b}{a} = \frac{\frac{26}{3}}{\frac{16}{3}} = \frac{26}{16} = \frac{13}{8}.

Final answer

138\frac{13}{8}

NEXT STEPS
Final Takeaways from Your Teacher

  • Review the distinction between the product rule (zm⋅zn=zm+nz^m \cdot z^n = z^{m+n}) and the quotient rule (zmzn=zm−n\frac{z^m}{z^n} = z^{m-n}).

  • Explicitly write the minus sign with parentheses when subtracting negative exponents in denominators.

  • Recalculate aa and bb using the quotient rule to confirm the correct ratio of 138\frac{13}{8}.

OverviewACEpath Teacher's Diagnosis

03 The student miscalculates the reciprocal ratio as a/b instead of b/a.

Sample Student Work

Reviewed by ACEpath Teacher
Step-by-Step Review
Walk Through It with Your Teacher
Incorrect
  1. Step 1

    Numerator: \text{Numerator: } x^{4/3} ⋅\cdot y^{-2/3} ⋅\cdot x^2 ⋅\cdot y^{-12} = x^{10/3} ⋅\cdot y^{-38/3}   ⟹  \implies Denominator: \text{Denominator: } x^{-2} ⋅\cdot y^{10/3}

  2. Step 2

    a = 103\frac{10}{3} - (-2) = 163\frac{16}{3}   ⟹  \implies b = -383\frac{38}{3} - 103\frac{10}{3} = -16

  3. Step 3

    ba=163−16=−13\frac{b}{a} = \frac{\frac{16}{3}}{-16} = -\frac{1}{3}

    Correct

    ba\frac{b}{a} = −16163\frac{-16}{\frac{16}{3}} = -16 ⋅\cdot 316\frac{3}{16} = -3

    You placed a=163a = \frac{16}{3} in the numerator and b=−16b = -16 in the denominator, which calculated ab\frac{a}{b} rather than the requested ratio ba\frac{b}{a}.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly applied the power of a product rule and rational exponent rules to simplify the numerator and denominator.

You correctly used the quotient rule for exponents by subtracting negative powers without making sign errors.

You arrived at the exact correct values for both individual exponents, aa and bb.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Inverted Ratio Substitution

You accidentally reversed the numerator and denominator when setting up the ratio, evaluating ab\frac{a}{b} instead of ba\frac{b}{a}.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Losing track of variable assignments when substituting into a fraction containing compound rational numbers leads to calculating the reciprocal of the target expression.

REASONING PATH
Your ACEpath teacher’s suggested approach
Check Variable Placement in Ratios

  • For any requested ratio topbottom\frac{\text{top}}{\text{bottom}}, identify the corresponding variable for each position before writing values.

  • Write the labeled skeleton ba=[][]\frac{b}{a} = \frac{[\quad]}{[\quad]} and explicitly insert the value of bb on top and aa on the bottom.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Identify the calculated values: a=163a = \frac{16}{3} and b=−16b = -16.

  2. Set up the ratio ba\frac{b}{a} with bb in the numerator and aa in the denominator: ba=−16163\frac{b}{a} = \frac{-16}{\frac{16}{3}}.

  3. Multiply by the reciprocal of the denominator: −16⋅316=−3-16 \cdot \frac{3}{16} = -3.

Final answer

-3

NEXT STEPS
Final Takeaways from Your Teacher

  • Before substituting numerical values into a fraction, write out the formula skeleton with variable names clearly visible.

  • Perform a quick mental sanity check: if ∣b∣>∣a∣|b| > |a|, then ∣ba∣\left|\frac{b}{a}\right| must be greater than 11.

Lock it in

Question

What is the difference between (xa)b(x^a)^b and xa⋅xbx^a \cdot x^b?

Answer

(xa)b=xa⋅b(x^a)^b = x^{a \cdot b} (multiply exponents for a power of a power). xa⋅xb=xa+bx^a \cdot x^b = x^{a+b} (add exponents when multiplying like bases).

Similar question

01

Simplify the expression (x−2y4)−1/2x3y−1\frac{(x^{-2} y^4)^{-1/2}}{x^3 y^{-1}} to the form xpyqx^p y^q. What are pp and qq?

Answer

Numerator: (x−2)−1/2=x1(x^{-2})^{-1/2} = x^1, (y4)−1/2=y−2(y^4)^{-1/2} = y^{-2}. Combining with the denominator gives x1−3y−2−(−1)=x−2y−1x^{1 - 3} y^{-2 - (-1)} = x^{-2} y^{-1}. Thus, p=−2p = -2 and q=−1q = -1.

02

Given that (a1/3b−2)3(a−1b4)−1/2=ambn\frac{(a^{1/3} b^{-2})^3}{(a^{-1} b^4)^{-1/2}} = a^m b^n, determine the value of m+nm + n.

Answer

Numerator: a1/3⋅3b−2⋅3=a1b−6a^{1/3 \cdot 3} b^{-2 \cdot 3} = a^1 b^{-6}. Denominator: a(−1)(−1/2)b4(−1/2)=a1/2b−2a^{(-1)(-1/2)} b^{4(-1/2)} = a^{1/2} b^{-2}. Subtract exponents: m=1−12=12m = 1 - \frac{1}{2} = \frac{1}{2}, and n=−6−(−2)=−4n = -6 - (-2) = -4. Therefore, m+n=12−4=−72m + n = \frac{1}{2} - 4 = -\frac{7}{2}.

03

For positive real numbers uu and vv, the expression u6v−4⋅(u−2/3v1/2)−6(u4v−8)1/4\frac{\sqrt{u^6 v^{-4}} \cdot (u^{-2/3} v^{1/2})^{-6}}{(u^4 v^{-8})^{1/4}} simplifies to urvsu^r v^s. Find the value of sr\frac{s}{r}.

Answer

First factor: (u6v−4)1/2=u3v−2(u^6 v^{-4})^{1/2} = u^3 v^{-2}. Second factor: u(−2/3)(−6)v(1/2)(−6)=u4v−3u^{(-2/3)(-6)} v^{(1/2)(-6)} = u^4 v^{-3}. Denominator: u4(1/4)v−8(1/4)=u1v−2u^{4(1/4)} v^{-8(1/4)} = u^1 v^{-2}. Combining: r=3+4−1=6r = 3 + 4 - 1 = 6, and s=−2+(−3)−(−2)=−3s = -2 + (-3) - (-2) = -3. Thus, sr=−36=−12\frac{s}{r} = \frac{-3}{6} = -\frac{1}{2}.

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