Resources

Evaluate and compare the signs and exact values of the following three expressions involving negative bases and rational exponents: A=−642/3A = -64^{2/3}, B=(−64)2/3B = (-64)^{2/3}, and C=(−64)−2/3C = (-64)^{-2/3}. Express your final answer as an ordered triple (A,B,C)(A, B, C) of simplified rational numbers.

Answer and steps

Final answer

(−16,16,116)\left(-16, 16, \frac{1}{16}\right)

Steps

01

Evaluate A=−642/3A = -64^{2/3}

To evaluate each expression, apply the rules for order of operations, rational exponents, and negative exponents step by step.

By the standard order of operations, an exponent takes precedence over a preceding negative sign unless parentheses explicitly group them together. Therefore, the base is 6464, not −64-64: A=−(642/3)A = -(64^{2/3})

Using the rational exponent property am/n=(an)ma^{m/n} = (\sqrt[n]{a})^m: 642/3=(643)2=42=1664^{2/3} = (\sqrt[3]{64})^2 = 4^2 = 16

Applying the negative sign gives: A=−16A = -16

Common Pitfall: Do not confuse −ap-a^p with (−a)p(-a)^p. In −642/3-64^{2/3}, only the number 6464 is raised to the power 23\frac{2}{3}.

02

Evaluate B=(−64)2/3B = (-64)^{2/3}

Here, the parentheses indicate that the entire negative number −64-64 is the base: B=(−64)2/3=(−643)2B = (-64)^{2/3} = (\sqrt[3]{-64})^2

  1. First, take the real cube root of the negative base: −643=−4since (−4)3=−64\sqrt[3]{-64} = -4 \quad \text{since } (-4)^3 = -64
  2. Next, square the resulting value: (−4)2=16(-4)^2 = 16

Thus: B=16B = 16

03

Evaluate C=(−64)−2/3C = (-64)^{-2/3}

A negative exponent indicates the reciprocal of the base raised to the positive power, defined by x−k=1xkx^{-k} = \frac{1}{x^k}: C=1(−64)2/3C = \frac{1}{(-64)^{2/3}}

From Step 2, we already know that (−64)2/3=16(-64)^{2/3} = 16: C=116C = \frac{1}{16}

Conclusion

Combining the results into an ordered triple: (A,B,C)=(−16,16,116)(A, B, C) = \left(-16, 16, \frac{1}{16}\right)

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student confuses -64^(2/3) with (-64)^(2/3), incorrectly absorbing the negative sign into the base for A to conclude that A = 16.

Sample Student Work

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

    A = -64^{2/3} = (-64)^{2/3} = 16

    Correct

    A = -(64^{2/3}) = -(643\sqrt[3]{64})^2 = -(4^2) = -16

    In the expression −642/3-64^{2/3}, there are no parentheses around −64-64. By the order of operations, exponentiation takes precedence over negation, so only 6464 is raised to the 23\frac{2}{3} power: A=−(642/3)=−16A = -(64^{2/3}) = -16.

  2. Step 2

    B = (-64)^{2/3} = ((-64)^{1/3})^2 = (-4)^2 = 16

  3. Step 3

    C = (-64)^{-2/3} = 1(−64)2/3\frac{1}{(-64)^{2/3}} = 116\frac{1}{16}

  4. Step 4

    (A,B,C)=(16,16,116)(A, B, C) = \left(16, 16, \frac{1}{16}\right)

    Correct

    (A,B,C)=(−16,16,116)(A, B, C) = \left(-16, 16, \frac{1}{16}\right)

    This final triple is affected solely by the incorrect sign found for AA in Step 1.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly evaluated the fractional power of a negative base enclosed in parentheses for expression BB.

You correctly applied the negative exponent rule to rewrite CC as a reciprocal.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Order of operations with negation and exponents

You treated −642/3-64^{2/3} as if it had parentheses around the base, setting it equal to (−64)2/3(-64)^{2/3}.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Confusing −xn-x^n with (−x)n(-x)^n. Exponentiation has higher precedence than unary negation, meaning −xn=−(xn)-x^n = -(x^n) unless parentheses explicitly group the negative sign with the base.

REASONING PATH
Your ACEpath teacher’s suggested approach
Check Parentheses First

  • −bp=−(bp)-b^p = -(b^p), whereas (−b)p(-b)^p raises the full signed value −b-b to the power pp.

  • If no parentheses enclose the negative sign, evaluate the exponent on the positive base first, then apply the negative sign at the end.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Identify that the exponent applies only to 6464: A=−(642/3)A = -(64^{2/3}).

  2. Evaluate the rational exponent: 642/3=(643)2=42=1664^{2/3} = (\sqrt[3]{64})^2 = 4^2 = 16.

  3. Apply the leading negative sign: A=−16A = -16.

  4. Combine the values into the ordered triple: (A,B,C)=(−16,16,116)(A, B, C) = \left(-16, 16, \frac{1}{16}\right).

Final answer

(A,B,C)=(−16,16,116)(A, B, C) = \left(-16, 16, \frac{1}{16}\right)

NEXT STEPS
Final Takeaways from Your Teacher

  • Pay special attention to whether a negative sign is inside or outside parentheses before computing any power.

  • Practice rewriting −am/n-a^{m/n} as −(am/n)-(a^{m/n}) as your very first step to avoid accidentally absorbing the minus sign into the base.

OverviewACEpath Teacher's Diagnosis

02 The student assumes that squaring a negative base inside a rational exponent maintains a negative sign, improperly obtaining B = -16 and C = -1/16.

Sample Student Work

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

    A=−(641/3)2=−(42)=−16A = -(64^{1/3})^2 = -(4^2) = -16

  2. Step 2

    B=((−64)1/3)2=(−4)2=−16B = ((-64)^{1/3})^2 = (-4)^2 = -16

    Correct

    B=((−64)1/3)2=(−4)2=16B = ((-64)^{1/3})^2 = (-4)^2 = 16

    You correctly found the cube root −643=−4\sqrt[3]{-64} = -4, but squaring a negative number inside parentheses produces a positive result: (−4)2=(−4)×(−4)=16(-4)^2 = (-4) \times (-4) = 16.

  3. Step 3

    C=1B=1(−16)=−116C = \frac{1}{B} = \frac{1}{(-16)} = -\frac{1}{16}

    Correct

    C=1B=116C = \frac{1}{B} = \frac{1}{16}

    Your reciprocal step C=1BC = \frac{1}{B} was mathematically sound, but it inherited the incorrect negative value of BB from Step 2.

  4. Step 4

    (A,B,C)=(−16,−16,−116)(A, B, C) = \left(-16, -16, -\frac{1}{16}\right)

    Correct

    (A,B,C)=(−16,16,116)(A, B, C) = \left(-16, 16, \frac{1}{16}\right)

    This ordered triple correctly gathered your computed values, but reflects the sign error made in Step 2.

WHAT YOU DID WELL
Compliments from Your Teacher

You distinguished −642/3-64^{2/3} from (−64)2/3(-64)^{2/3} correctly by isolating the leading negative sign in AA.

You calculated the odd cube root −643=−4\sqrt[3]{-64} = -4 correctly.

You recognized that a negative exponent denotes the reciprocal of the positive power.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Sign error when squaring a negative base

Evaluating (−4)2(-4)^2 as −16-16 instead of +16+16.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

You conflated −x2-x^2 with (−x)2(-x)^2, treating the square of a negative number as retaining its negative sign rather than multiplying two negative factors to yield a positive product.

REASONING PATH
Your ACEpath teacher’s suggested approach
Even powers of negative bases

  • For any real number xx and even integer 2k2k, (−x)2k>0(-x)^{2k} > 0 when x≠0x \neq 0.

  • Whenever you see parentheses around a negative number raised to an even power, remind yourself that negative times negative equals positive: (−a)2=+a2(-a)^2 = +a^2.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Compute the square: (−4)2=(−4)×(−4)=16(-4)^2 = (-4) \times (-4) = 16.

  2. Substitute the correct positive value of BB into expression CC: C=1B=116C = \frac{1}{B} = \frac{1}{16}.

  3. Combine the three evaluated values into the ordered triple: (A,B,C)=(−16,16,116)(A, B, C) = \left(-16, 16, \frac{1}{16}\right).

Final answer

(−16,16,116)\left(-16, 16, \frac{1}{16}\right)

NEXT STEPS
Final Takeaways from Your Teacher

  • Review the distinction between −a2-a^2 and (−a)2(-a)^2 with explicit expansion: −a2=−(a⋅a)-a^2 = -(a \cdot a) versus (−a)2=(−a)⋅(−a)(-a)^2 = (-a) \cdot (-a).

  • Practice evaluating expressions with fractional exponents by separating the root (denominator) and power (numerator) explicitly.

OverviewACEpath Teacher's Diagnosis

03 The student misinterprets the negative exponent in C as making the value negative rather than taking the reciprocal, yielding C = -16.

Sample Student Work

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

    A = -(64^{23\frac{2}{3}}) = -(4^2) = -16

  2. Step 2

    B = ((-64)^{13\frac{1}{3}})^2 = (-4)^2 = 16

  3. Step 3

    C = (-64)^{-23\frac{2}{3}} = -((-64)^{23\frac{2}{3}}) = -16

    Correct

    C = 1(−64)2/3\frac{1}{(-64)^{2/3}} = 116\frac{1}{16}

    A negative exponent denotes a reciprocal (x−n=1xnx^{-n} = \frac{1}{x^n}), not the additive inverse (negation) of the power. Therefore, (−64)−2/3=1(−64)2/3=116(-64)^{-2/3} = \frac{1}{(-64)^{2/3}} = \frac{1}{16}.

  4. Step 4

    (A, B, C) = (-16, 16, -16)

    Correct

    (A,B,C)=(−16,16,116)(A, B, C) = \left(-16, 16, \frac{1}{16}\right)

    This final ordered triple followed directly from the incorrect value obtained for CC in Step 3.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly applied the order of operations to distinguish −ap-a^p from (−a)p(-a)^p.

You evaluated the fractional exponent of a negative base correctly in expression BB by taking the cube root first and squaring the result.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Negative exponent misinterpreted as sign negation

You pulled the negative sign out from the exponent to the front of the expression, writing x−k=−(xk)x^{-k} = -(x^k) instead of inverting the base to 1xk\frac{1}{x^k}.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Confusing the role of a negative sign in the exponent (reciprocal operation) with a negative coefficient or factor (additive inverse).

REASONING PATH
Your ACEpath teacher’s suggested approach
Negative Exponent Rule

  • For any non-zero base xx and real exponent kk, x−k=1xkx^{-k} = \frac{1}{x^k}.

  • Whenever you see a negative exponent, rewrite the expression as a fraction with 11 in the numerator and the positive exponent in the denominator before evaluating.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Apply the negative exponent property: C=(−64)−2/3=1(−64)2/3C = (-64)^{-2/3} = \frac{1}{(-64)^{2/3}}.

  2. Substitute the value of (−64)2/3=16(-64)^{2/3} = 16 found in Step 2: C=116C = \frac{1}{16}.

  3. Assemble the corrected components into the ordered triple: (A,B,C)=(−16,16,116)(A, B, C) = \left(-16, 16, \frac{1}{16}\right).

Final answer

(−16,16,116)\left(-16, 16, \frac{1}{16}\right)

NEXT STEPS
Final Takeaways from Your Teacher

  • Practice converting negative exponents into reciprocals before performing any arithmetic.

  • Remember that an exponent only changes the sign of the overall result if it represents an odd power applied to a negative base, never simply because the exponent itself is negative.

Lock it in

Question

What is the structural difference between −xn-x^n and (−x)n(-x)^n?

Answer

In −xn-x^n, only xx is raised to the power nn, giving −(xn)-(x^n). In (−x)n(-x)^n, parentheses group −x-x as the entire base.

Similar question

01

Evaluate and simplify the exact value of −84/3-8^{4/3}.

Answer

−16-16. By the order of operations, −84/3=−(84/3)=−(83)4=−(24)=−16-8^{4/3} = -(8^{4/3}) = -(\sqrt[3]{8})^4 = -(2^4) = -16.

02

Evaluate both P=−323/5P = -32^{3/5} and Q=(−32)3/5Q = (-32)^{3/5}. Do they have the same numerical value?

Answer

Yes, both equal −8-8. For PP, −323/5=−(323/5)=−(23)=−8-32^{3/5} = -(32^{3/5}) = -(2^3) = -8. For QQ, (−32)3/5=(−325)3=(−2)3=−8(-32)^{3/5} = (\sqrt[5]{-32})^3 = (-2)^3 = -8. Their operations differ, but an odd power preserves the negative sign.

03

Evaluate the ordered pair (X,Y)=(−813/4, (−27)−4/3)(X, Y) = \left(-81^{3/4},\, (-27)^{-4/3}\right) as simplified rational numbers.

Answer

(−27,181)\left(-27, \frac{1}{81}\right). For XX, −813/4=−(813/4)=−((814)3)=−(33)=−27-81^{3/4} = -(81^{3/4}) = -((\sqrt[4]{81})^3) = -(3^3) = -27. For YY, (−27)−4/3=1((−27)1/3)4=1(−3)4=181(-27)^{-4/3} = \frac{1}{((-27)^{1/3})^4} = \frac{1}{(-3)^4} = \frac{1}{81}.

Loading workspace…