Simplify the algebraic expression by eliminating all negative and zero exponents: $\frac{(16 x^{-4} y^6 z^0)^{3/2}}{2 x^{-8} y^{-3} z^{-2}}$. Write the final result in terms of positive rational exponents only, with all coefficients fully simplified.

Simplify the algebraic expression by eliminating all negative and zero exponents: $\frac{(16 x^{-4} y^6 z^0)^{3/2}}{2 x^{-8} y^{-3} z^{-2}}$ . Write the final result in terms of positive rational exponents only, with all coefficients fully simplified.

Final answer

$32 x^2 y^{12} z^2$

Steps

Simplify the numerator using the power of a product rule

To simplify the expression, apply the rules of exponents step by step.

Recall that $(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n$ and $(a^m)^n = a^{m \cdot n}$ . Also, any nonzero base raised to the power of $0$ equals $1$ ( $z^0 = 1$ ).

Distribute the rational exponent $\frac{3}{2}$ to each factor inside the parentheses:

$(16 x^{-4} y^6 z^0)^{3/2} = 16^{3/2} \cdot (x^{-4})^{3/2} \cdot (y^6)^{3/2} \cdot (z^0)^{3/2}$

Evaluate each term individually:

Coefficient: $16^{3/2} = (\sqrt{16})^3 = 4^3 = 64$
$x$ -term: $(x^{-4})^{3/2} = x^{-4 \cdot \frac{3}{2}} = x^{-6}$
$y$ -term: $(y^6)^{3/2} = y^{6 \cdot \frac{3}{2}} = y^9$
$z$ -term: $(z^0)^{3/2} = 1^{3/2} = 1$

Multiplying these together gives the simplified numerator: $64 x^{-6} y^9$

Set up the full fraction

Substitute the simplified numerator back into the original expression:

$\frac{64 x^{-6} y^9}{2 x^{-8} y^{-3} z^{-2}}$

Simplify by grouping coefficients and like bases

Use the quotient rule of exponents, $\frac{a^m}{a^n} = a^{m - n}$ , for each base:

Coefficients: $\frac{64}{2} = 32$
Base $x$ : $\frac{x^{-6}}{x^{-8}} = x^{-6 - (-8)} = x^{-6 + 8} = x^2$
Base $y$ : $\frac{y^9}{y^{-3}} = y^{9 - (-3)} = y^{9 + 3} = y^{12}$
Base $z$ : $\frac{1}{z^{-2}} = z^{-(-2)} = z^2$

Common Pitfall: When applying the quotient rule with negative exponents, remember that subtracting a negative number becomes addition: $-6 - (-8) = 2$ , not $-14$ .

Combine the simplified terms

Combine all simplified parts into a single product:

$32 x^2 y^{12} z^2$

All exponents are positive and the expression is fully simplified.

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student multiplies the base coefficient by the fractional exponent instead of raising it to the power, calculating 16 * (3/2) = 24 instead of 16^(3/2) = 64.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
$\frac{16 \cdot \frac{3}{2} x^{-6} y^9 (1)}{2 x^{-8} y^{-3} z^{-2}}$ $\implies$ $\frac{24 x^{-6} y^9}{2 x^{-8} y^{-3} z^{-2}}$
Correct
$\frac{16^{3/2} x^{-6} y^9 (1)}{2 x^{-8} y^{-3} z^{-2}}$ = $\frac{64 x^{-6} y^9}{2 x^{-8} y^{-3} z^{-2}}$
When applying the power of a product rule $(a \cdot b)^n = a^n b^n$ , the numerical coefficient must be raised to the exponent $16^{3/2} = (\sqrt{16})^3 = 4^3 = 64$ , rather than multiplied by the exponent as $16 \cdot \frac{3}{2} = 24$ .
Step 2
12 x^{-6 - (-8)} y^{9 - (-3)} z^{0 - (-2)}
Correct
32 x^{-6 - (-8)} y^{9 - (-3)} z^{0 - (-2)}
You correctly set up the quotient rule $\frac{a^m}{a^n} = a^{m-n}$ for each variable base. The coefficient $12$ is an arithmetic consequence of the earlier error ( $\frac{24}{2} = 12$ instead of $\frac{64}{2} = 32$ ).
Step 3
12 x^2 y^{12} z^2
Correct
32 x^2 y^{12} z^2
Your subtraction of negative exponents is completely correct ( $-6 - (-8) = 2$ , $9 - (-3) = 12$ , and $0 - (-2) = 2$ ). The final term reflects the propagated coefficient error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly distributed the outer exponent to all variable powers using the power rule $(x^m)^n = x^{m \cdot n}$ .

You correctly evaluated $z^0 = 1$ .

You correctly applied the quotient rule $\frac{a^m}{a^n} = a^{m-n}$ and handled subtracting negative exponents without any sign errors.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Coefficient exponentiation confused with multiplication

No; this represents a conceptual slip regarding how power-of-a-product applies to the numerical base versus variable exponents.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Conflating the operation on variable exponents (multiplying the powers, $m \cdot n$ ) with the operation on the base coefficient (which must be raised to the power, $c^n$ , not multiplied by $n$ ).

REASONING PATH

Your ACEpath teacher’s suggested approach

Power of a Product with Coefficients: $(c \cdot x^k)^n = c^n \cdot x^{k \cdot n}$
Treat numerical coefficients as bases with an implicit exponent of $1$ : $16 = 16^1$ , so $(16^1)^{3/2} = 16^{3/2} = 64$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Evaluate the coefficient with the outer exponent: $16^{3/2} = (\sqrt{16})^3 = 4^3 = 64$ .
Form the corrected simplified numerator: $64 x^{-6} y^9$ .
Divide by the denominator $2 x^{-8} y^{-3} z^{-2}$ : $\frac{64}{2} x^{-6 - (-8)} y^{9 - (-3)} z^{0 - (-2)} = 32 x^2 y^{12} z^2$ .

Final answer

32 x^2 y^{12} z^2

NEXT STEPS

Final Takeaways from Your Teacher

Always write an explicit exponent of $1$ on numbers inside parentheses, such as $(16^1 x^{-4})^{3/2}$ , to remind yourself that exponents multiply exponents: $1 \cdot \frac{3}{2} = \frac{3}{2}$ , giving $16^{3/2}$ .
Practice evaluating rational exponents separately on scrap paper before combining them with algebraic terms.

OverviewACEpath Teacher's Diagnosis

02 The student subtracts negative exponents incorrectly by adding them or dropping double negatives, calculating -6 - (-8) as -6 - 8 = -14 and 9 - (-3) as 9 - 3 = 6.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{64 x^{-6} y^9}{2 x^{-8} y^{-3} z^{-2}}$
Step 2
32 x^{-6-8} y^{9-3} z^{-2} $\implies$ 32 x^{-14} y^6 z^{-2}
Correct
32 x^{-6 - (-8)} y^{9 - (-3)} z^{0 - (-2)} = 32 x^2 y^{12} z^2
When dividing like bases using $\frac{a^m}{a^n} = a^{m - n}$ , subtracting a negative denominator exponent requires parentheses: $-6 - (-8) = -6 + 8 = 2$ , and $9 - (-3) = 9 + 3 = 12$ . You wrote $-6 - 8$ and $9 - 3$ , which dropped the double negatives.
Step 3
$\frac{32 y^6}{x^{14} z^2}$
Correct
32 x^2 y^{12} z^2
You converted your negative exponents to positive exponents in the denominator accurately, but the values were carried over from the subtraction error in Step 2.

WHAT YOU DID WELL

Compliments from Your Teacher

You evaluated $16^{3/2} = (\sqrt{16})^3 = 64$ correctly.

You applied the power of a power rule $(a^m)^n = a^{m \cdot n}$ to each variable in the numerator correctly.

You simplified the numerical coefficients $\frac{64}{2} = 32$ without error.

You correctly rewrote negative exponents into denominators in your final step.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign error when subtracting negative exponents

You treated subtracting a negative exponent as subtracting a positive exponent, writing $-6 - 8$ instead of $-6 - (-8)$ and $9 - 3$ instead of $9 - (-3)$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking the double negative in the quotient rule formula $\frac{a^m}{a^n} = a^{m - n}$ when the denominator power $n$ is already negative, leading to omission of parentheses and incorrect signs.

REASONING PATH

Your ACEpath teacher’s suggested approach

Use parentheses when applying the quotient rule: $\frac{a^m}{a^n} = a^{m - (n)}$
Always write the denominator exponent inside parentheses before simplifying: $x^{m - (n)}$ .
Convert double negatives to addition: m - (-n) = m + n
Check whether the bottom power has a minus sign; if it does, subtracting it turns the operation into addition.

CORRECTED PATH

ACEpath Teacher's Demonstration

Divide coefficients: $\frac{64}{2} = 32$ .
Subtract denominator exponents from numerator exponents using parentheses: $x^{-6 - (-8)} = x^{-6 + 8} = x^2$ .
Subtract the $y$ exponent: $y^{9 - (-3)} = y^{9 + 3} = y^{12}$ .
Bring $z^{-2}$ from the denominator to the numerator: $\frac{1}{z^{-2}} = z^{-(-2)} = z^2$ .

Final answer

32 x^2 y^{12} z^2

NEXT STEPS

Final Takeaways from Your Teacher

Practice writing the quotient rule with explicit parentheses around negative exponents, such as $x^{m - (n)}$ .
Alternative method: Move negative power factors across the fraction bar first (e.g., $\frac{x^{-6}}{x^{-8}} = \frac{x^8}{x^6} = x^{8 - 6} = x^2$ ) to eliminate negative signs before subtracting.

OverviewACEpath Teacher's Diagnosis

03 The student adds exponents instead of multiplying them when applying the power of a power rule to the numerator variables.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{64 x^{-4 + 3/2} y^{6 + 3/2} z^0}{2 x^{-8} y^{-3} z^{-2}}$ $\implies$ $\frac{64 x^{-5/2} y^{15/2}}{2 x^{-8} y^{-3} z^{-2}}$
Correct
$\frac{64 x^{-4 \cdot (3/2)} y^{6 \cdot (3/2)}}{2 x^{-8} y^{-3} z^{-2}}$ = $\frac{64 x^{-6} y^9}{2 x^{-8} y^{-3} z^{-2}}$
When raising a power to another power, you must multiply the exponents using the rule $(a^m)^n = a^{m \cdot n}$ . Here, you added them as $-4 + \frac{3}{2}$ and $6 + \frac{3}{2}$ instead of multiplying $-4 \cdot \frac{3}{2} = -6$ and $6 \cdot \frac{3}{2} = 9$ .
Step 2
32 x^{-5/2 - (-8)} y^{15/2 - (-3)} z^2
Correct
32 x^{-6 - (-8)} y^{9 - (-3)} z^2
Your application of the quotient rule $\frac{a^m}{a^n} = a^{m-n}$ and division of the numerical coefficients $\frac{64}{2} = 32$ were completely correct, but the values were affected by the earlier power-of-a-power mistake.
Step 3
32 x^{11/2} y^{21/2} z^2
Correct
32 x^2 y^{12} z^2
Your arithmetic on the exponents was carried out correctly from your previous line, but the final powers remain incorrect due to the initial multiplication error in the numerator.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly evaluated the numerical fractional power $16^{3/2} = (\sqrt{16})^3 = 64$ .

You handled $z^0 = 1$ and correctly applied the negative exponent rule to move $z^{-2}$ from the denominator to $z^2$ in the numerator.

You correctly subtracted the denominator exponents when applying the quotient rule.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Power of a power rule confused with product rule

You wrote $-4 + \frac{3}{2}$ and $6 + \frac{3}{2}$ across the variable terms instead of multiplying the inner and outer exponents.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the product of powers rule $a^m \cdot a^n = a^{m+n}$ with the power of a power rule $(a^m)^n = a^{m \cdot n}$ . When an expression inside parentheses is raised to an outer exponent, each exponent inside must be multiplied by the outer exponent.

REASONING PATH

Your ACEpath teacher’s suggested approach

Power of a Power Rule: $(a^m)^n = a^{m \cdot n}$
Multiply the inner exponent by the outer exponent whenever an exponential term is raised to a power.
Product of Powers Rule: $a^m \cdot a^n = a^{m+n}$
Only add exponents when multiplying two identical bases together.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute the outer exponent $\frac{3}{2}$ by multiplying exponents: $(x^{-4})^{3/2} = x^{-4 \cdot \frac{3}{2}} = x^{-6}$ , and $(y^6)^{3/2} = y^{6 \cdot \frac{3}{2}} = y^9$ .
Rewrite the full fraction: $\frac{64 x^{-6} y^9}{2 x^{-8} y^{-3} z^{-2}}$ .
Apply the quotient rule to like bases: $\frac{64}{2} = 32$ , $x^{-6 - (-8)} = x^2$ , $y^{9 - (-3)} = y^{12}$ , and $\frac{1}{z^{-2}} = z^2$ .

Final answer

32 x^2 y^{12} z^2

NEXT STEPS

Final Takeaways from Your Teacher

Review the distinction between $(x^a)^b = x^{a \cdot b}$ and $x^a \cdot x^b = x^{a+b}$ .
Write out an explicit multiplication dot, such as $x^{-4 \cdot 3/2}$ , whenever distributing an outer exponent over parentheses.
Practice simplifying expressions that combine fractional outer powers with negative base exponents.

Lock it in

Question

What is the rule for distributing an exponent to a numerical coefficient in an expression like $(c \cdot x^k)^n$ ?

Answer

$(c \cdot x^k)^n = c^n \cdot x^{k \cdot n}$ . The coefficient is a base raised to the power $n$ , not multiplied by $n$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student multiplies the base coefficient by the fractional exponent instead of raising it to the power, calculating 16 * (3/2) = 24 instead of 16^(3/2) = 64.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student subtracts negative exponents incorrectly by adding them or dropping double negatives, calculating -6 - (-8) as -6 - 8 = -14 and 9 - (-3) as 9 - 3 = 6.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student adds exponents instead of multiplying them when applying the power of a power rule to the numerator variables.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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