Basic Math Examples

${(\frac{27 p^{2} y^{3} z^{- 3}}{8 p^{5} y^{- 2} z^{2}})}^{- 4}$

Step 1

Move $z^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{27 p^{2} y^{3}}{8 p^{5} y^{- 2} z^{2} z^{3}})}^{- 4}$

Step 2

Multiply $z^{2}$ by $z^{3}$ by adding the exponents.

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Step 2.1

Move $z^{3}$ .

${(\frac{27 p^{2} y^{3}}{8 p^{5} y^{- 2} (z^{3} z^{2})})}^{- 4}$

Step 2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{27 p^{2} y^{3}}{8 p^{5} y^{- 2} z^{3 + 2}})}^{- 4}$

Step 2.3

Add $3$ and $2$ .

${(\frac{27 p^{2} y^{3}}{8 p^{5} y^{- 2} z^{5}})}^{- 4}$

Step 3

Move $y^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{27 p^{2} y^{3} y^{2}}{8 p^{5} z^{5}})}^{- 4}$

Step 4

Multiply $y^{3}$ by $y^{2}$ by adding the exponents.

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Step 4.1

Move $y^{2}$ .

${(\frac{27 p^{2} (y^{2} y^{3})}{8 p^{5} z^{5}})}^{- 4}$

Step 4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{27 p^{2} y^{2 + 3}}{8 p^{5} z^{5}})}^{- 4}$

Step 4.3

Add $2$ and $3$ .

${(\frac{27 p^{2} y^{5}}{8 p^{5} z^{5}})}^{- 4}$

Step 5

Cancel the common factor of $p^{2}$ and $p^{5}$ .

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Step 5.1

Factor $p^{2}$ out of $27 p^{2} y^{5}$ .

${(\frac{p^{2} (27 y^{5})}{8 p^{5} z^{5}})}^{- 4}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $p^{2}$ out of $8 p^{5} z^{5}$ .

${(\frac{p^{2} (27 y^{5})}{p^{2} (8 p^{3} z^{5})})}^{- 4}$

Step 5.2.2

Cancel the common factor.

${(\frac{p^{2} (27 y^{5})}{p^{2} (8 p^{3} z^{5})})}^{- 4}$

Step 5.2.3

Rewrite the expression.

${(\frac{27 y^{5}}{8 p^{3} z^{5}})}^{- 4}$

Step 6

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{8 p^{3} z^{5}}{27 y^{5}})}^{4}$

Step 7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1

Apply the product rule to $\frac{8 p^{3} z^{5}}{27 y^{5}}$ .

$\frac{{(8 p^{3} z^{5})}^{4}}{{(27 y^{5})}^{4}}$

Step 7.2

Apply the product rule to $8 p^{3} z^{5}$ .

$\frac{{(8 p^{3})}^{4} {(z^{5})}^{4}}{{(27 y^{5})}^{4}}$

Step 7.3

Apply the product rule to $8 p^{3}$ .

$\frac{8^{4} {(p^{3})}^{4} {(z^{5})}^{4}}{{(27 y^{5})}^{4}}$

Step 7.4

Apply the product rule to $27 y^{5}$ .

$\frac{8^{4} {(p^{3})}^{4} {(z^{5})}^{4}}{27^{4} {(y^{5})}^{4}}$

Step 8

Simplify the numerator.

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Step 8.1

Raise $8$ to the power of $4$ .

$\frac{4096 {(p^{3})}^{4} {(z^{5})}^{4}}{27^{4} {(y^{5})}^{4}}$

Step 8.2

Multiply the exponents in ${(p^{3})}^{4}$ .

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Step 8.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4096 p^{3 \cdot 4} {(z^{5})}^{4}}{27^{4} {(y^{5})}^{4}}$

Step 8.2.2

Multiply $3$ by $4$ .

$\frac{4096 p^{12} {(z^{5})}^{4}}{27^{4} {(y^{5})}^{4}}$

Step 8.3

Multiply the exponents in ${(z^{5})}^{4}$ .

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Step 8.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4096 p^{12} z^{5 \cdot 4}}{27^{4} {(y^{5})}^{4}}$

Step 8.3.2

Multiply $5$ by $4$ .

$\frac{4096 p^{12} z^{20}}{27^{4} {(y^{5})}^{4}}$

Step 9

Simplify the denominator.

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Step 9.1

Raise $27$ to the power of $4$ .

$\frac{4096 p^{12} z^{20}}{531441 {(y^{5})}^{4}}$

Step 9.2

Multiply the exponents in ${(y^{5})}^{4}$ .

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Step 9.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4096 p^{12} z^{20}}{531441 y^{5 \cdot 4}}$

Step 9.2.2

Multiply $5$ by $4$ .

$\frac{4096 p^{12} z^{20}}{531441 y^{20}}$