Solved on Jan 18, 2024

Find the value of $125 z^{3} y^{12}$ .

STEP 1

Assumptions
1. We are given the expression $\left(5 z y^{4}\right)^{3}$ .
2. We need to simplify the expression by expanding the power over the product inside the parentheses.

STEP 2

Apply the power of a product rule, which states that $(abc)^n = a^n b^n c^n$ , where $a$ , $b$ , and $c$ are any real numbers and $n$ is a positive integer.

STEP 3

Identify the base components within the parentheses which are $5$ , $z$ , and $y^4$ .

STEP 4

Apply the power of a product rule to the expression $\left(5 z y^{4}\right)^{3}$ .
$\left(5 z y^{4}\right)^{3} = 5^3 z^3 \left(y^4\right)^3$

STEP 5

Calculate the power of the numerical coefficient $5^3$ .
$5^3 = 5 \times 5 \times 5 = 125$

STEP 6

Apply the power of a power rule to $\left(y^4\right)^3$ , which states that $(a^m)^n = a^{mn}$ .
$\left(y^4\right)^3 = y^{4 \times 3}$

STEP 7

Calculate the new exponent for $y$ .
$y^{4 \times 3} = y^{12}$

STEP 8

Combine the results from STEP_5 and STEP_7 with the power of $z$ .
$5^3 z^3 \left(y^4\right)^3 = 125 z^3 y^{12}$

STEP 9

Write the final simplified expression without parentheses.
$\left(5 z y^{4}\right)^{3} = 125 z^3 y^{12}$
This is the simplified form of the given expression.

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Find the value of $125 z^{3} y^{12}$ .

STEP 1

Assumptions
1. We are given the expression $\left(5 z y^{4}\right)^{3}$ .
2. We need to simplify the expression by expanding the power over the product inside the parentheses.

STEP 2

Apply the power of a product rule, which states that $(abc)^n = a^n b^n c^n$ , where $a$ , $b$ , and $c$ are any real numbers and $n$ is a positive integer.

STEP 3

Identify the base components within the parentheses which are $5$ , $z$ , and $y^4$ .

STEP 4

Apply the power of a product rule to the expression $\left(5 z y^{4}\right)^{3}$ .
$\left(5 z y^{4}\right)^{3} = 5^3 z^3 \left(y^4\right)^3$

STEP 5

Calculate the power of the numerical coefficient $5^3$ .
$5^3 = 5 \times 5 \times 5 = 125$

STEP 6

Apply the power of a power rule to $\left(y^4\right)^3$ , which states that $(a^m)^n = a^{mn}$ .
$\left(y^4\right)^3 = y^{4 \times 3}$

STEP 7

Calculate the new exponent for $y$ .
$y^{4 \times 3} = y^{12}$

STEP 8

Combine the results from STEP_5 and STEP_7 with the power of $z$ .
$5^3 z^3 \left(y^4\right)^3 = 125 z^3 y^{12}$

STEP 9

Write the final simplified expression without parentheses.
$\left(5 z y^{4}\right)^{3} = 125 z^3 y^{12}$
This is the simplified form of the given expression.

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Find the value of 125z3y12125 z^{3} y^{12}125z3y12.

STEP 1

Assumptions1. We are given the expression (5zy4)3\left(5 z y^{4}\right)^{3}(5zy4)3.2. We need to simplify the expression by expanding the power over the product inside the parentheses.

STEP 2

Apply the power of a product rule, which states that (abc)n=anbncn(abc)^n = a^n b^n c^n(abc)n=anbncn, where aaa, bbb, and ccc are any real numbers and nnn is a positive integer.

STEP 3

Identify the base components within the parentheses which are 555, zzz, and y4y^4y4.

STEP 4

Apply the power of a product rule to the expression (5zy4)3\left(5 z y^{4}\right)^{3}(5zy4)3.(5zy4)3=53z3(y4)3\left(5 z y^{4}\right)^{3} = 5^3 z^3 \left(y^4\right)^3(5zy4)3=53z3(y4)3

STEP 5

Calculate the power of the numerical coefficient 535^353.53=5×5×5=1255^3 = 5 \times 5 \times 5 = 12553=5×5×5=125

STEP 6

Apply the power of a power rule to (y4)3\left(y^4\right)^3(y4)3, which states that (am)n=amn(a^m)^n = a^{mn}(am)n=amn.(y4)3=y4×3\left(y^4\right)^3 = y^{4 \times 3}(y4)3=y4×3

STEP 7

Calculate the new exponent for yyy.y4×3=y12y^{4 \times 3} = y^{12}y4×3=y12

STEP 8

Combine the results from STEP_5 and STEP_7 with the power of zzz.53z3(y4)3=125z3y125^3 z^3 \left(y^4\right)^3 = 125 z^3 y^{12}53z3(y4)3=125z3y12

STEP 9

Write the final simplified expression without parentheses.(5zy4)3=125z3y12\left(5 z y^{4}\right)^{3} = 125 z^3 y^{12}(5zy4)3=125z3y12This is the simplified form of the given expression.

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Find the value of $125 z^{3} y^{12}$ .

Assumptions
1. We are given the expression $\left(5 z y^{4}\right)^{3}$ .
2. We need to simplify the expression by expanding the power over the product inside the parentheses.

Apply the power of a product rule, which states that $(abc)^n = a^n b^n c^n$ , where $a$ , $b$ , and $c$ are any real numbers and $n$ is a positive integer.

Identify the base components within the parentheses which are $5$ , $z$ , and $y^4$ .

Apply the power of a product rule to the expression $\left(5 z y^{4}\right)^{3}$ .
$\left(5 z y^{4}\right)^{3} = 5^3 z^3 \left(y^4\right)^3$

Calculate the power of the numerical coefficient $5^3$ .
$5^3 = 5 \times 5 \times 5 = 125$

Apply the power of a power rule to $\left(y^4\right)^3$ , which states that $(a^m)^n = a^{mn}$ .
$\left(y^4\right)^3 = y^{4 \times 3}$

Calculate the new exponent for $y$ .
$y^{4 \times 3} = y^{12}$

Combine the results from STEP_5 and STEP_7 with the power of $z$ .
$5^3 z^3 \left(y^4\right)^3 = 125 z^3 y^{12}$

Write the final simplified expression without parentheses.
$\left(5 z y^{4}\right)^{3} = 125 z^3 y^{12}$
This is the simplified form of the given expression.