Basic Math Examples

${(\frac{1}{4})}^{y + 2} = \sqrt[3]{8^{2 y - 1}}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt[3]{8^{2 y - 1}} = {(\frac{1}{4})}^{y + 2}$

Step 2

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{8^{2 y - 1}}}^{3} = {({(\frac{1}{4})}^{y + 2})}^{3}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{8^{2 y - 1}}$ as $8^{\frac{2 y - 1}{3}}$ .

${(8^{\frac{2 y - 1}{3}})}^{3} = {({(\frac{1}{4})}^{y + 2})}^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply the exponents in ${(8^{\frac{2 y - 1}{3}})}^{3}$ .

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Step 3.2.1.1

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

$8^{\frac{2 y - 1}{3} \cdot 3} = {({(\frac{1}{4})}^{y + 2})}^{3}$

Step 3.2.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.2.1

Cancel the common factor.

$8^{\frac{2 y - 1}{3} \cdot 3} = {({(\frac{1}{4})}^{y + 2})}^{3}$

Step 3.2.1.2.2

Rewrite the expression.

$8^{2 y - 1} = {({(\frac{1}{4})}^{y + 2})}^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${({(\frac{1}{4})}^{y + 2})}^{3}$ .

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Step 3.3.1.1

Apply the product rule to $\frac{1}{4}$ .

$8^{2 y - 1} = {(\frac{1^{y + 2}}{4^{y + 2}})}^{3}$

Step 3.3.1.2

One to any power is one.

$8^{2 y - 1} = {(\frac{1}{4^{y + 2}})}^{3}$

Step 3.3.1.3

Apply the product rule to $\frac{1}{4^{y + 2}}$ .

$8^{2 y - 1} = \frac{1^{3}}{{(4^{y + 2})}^{3}}$

Step 3.3.1.4

One to any power is one.

$8^{2 y - 1} = \frac{1}{{(4^{y + 2})}^{3}}$

Step 3.3.1.5

Multiply the exponents in ${(4^{y + 2})}^{3}$ .

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Step 3.3.1.5.1

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

$8^{2 y - 1} = \frac{1}{4^{(y + 2) \cdot 3}}$

Step 3.3.1.5.2

Apply the distributive property.

$8^{2 y - 1} = \frac{1}{4^{y \cdot 3 + 2 \cdot 3}}$

Step 3.3.1.5.3

Move $3$ to the left of $y$ .

$8^{2 y - 1} = \frac{1}{4^{3 \cdot y + 2 \cdot 3}}$

Step 3.3.1.5.4

Multiply $2$ by $3$ .

$8^{2 y - 1} = \frac{1}{4^{3 y + 6}}$

Step 4

Solve for $y$ .

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Step 4.1

Move $4^{3 y + 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$8^{2 y - 1} = 4^{- (3 y + 6)}$

Step 4.2

Create equivalent expressions in the equation that all have equal bases.

$2^{3 (2 y - 1)} = 2^{2 (- (3 y + 6))}$

Step 4.3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 (2 y - 1) = 2 (- (3 y + 6))$

Step 4.4

Solve for $y$ .

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Step 4.4.1

Simplify $3 (2 y - 1)$ .

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Step 4.4.1.1

Rewrite.

$0 + 0 + 3 (2 y - 1) = 2 (- (3 y + 6))$

Step 4.4.1.2

Simplify by adding zeros.

$3 (2 y - 1) = 2 (- (3 y + 6))$

Step 4.4.1.3

Apply the distributive property.

$3 (2 y) + 3 \cdot - 1 = 2 (- (3 y + 6))$

Step 4.4.1.4

Multiply.

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Step 4.4.1.4.1

Multiply $2$ by $3$ .

$6 y + 3 \cdot - 1 = 2 (- (3 y + 6))$

Step 4.4.1.4.2

Multiply $3$ by $- 1$ .

$6 y - 3 = 2 (- (3 y + 6))$

Step 4.4.2

Simplify $2 (- (3 y + 6))$ .

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Step 4.4.2.1

Apply the distributive property.

$6 y - 3 = 2 (- (3 y) - 1 \cdot 6)$

Step 4.4.2.2

Multiply.

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Step 4.4.2.2.1

Multiply $3$ by $- 1$ .

$6 y - 3 = 2 (- 3 y - 1 \cdot 6)$

Step 4.4.2.2.2

Multiply $- 1$ by $6$ .

$6 y - 3 = 2 (- 3 y - 6)$

Step 4.4.2.3

Apply the distributive property.

$6 y - 3 = 2 (- 3 y) + 2 \cdot - 6$

Step 4.4.2.4

Multiply.

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Step 4.4.2.4.1

Multiply $- 3$ by $2$ .

$6 y - 3 = - 6 y + 2 \cdot - 6$

Step 4.4.2.4.2

Multiply $2$ by $- 6$ .

$6 y - 3 = - 6 y - 12$

Step 4.4.3

Move all terms containing $y$ to the left side of the equation.

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Step 4.4.3.1

Add $6 y$ to both sides of the equation.

$6 y - 3 + 6 y = - 12$

Step 4.4.3.2

Add $6 y$ and $6 y$ .

$12 y - 3 = - 12$

Step 4.4.4

Move all terms not containing $y$ to the right side of the equation.

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Step 4.4.4.1

Add $3$ to both sides of the equation.

$12 y = - 12 + 3$

Step 4.4.4.2

Add $- 12$ and $3$ .

$12 y = - 9$

Step 4.4.5

Divide each term in $12 y = - 9$ by $12$ and simplify.

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Step 4.4.5.1

Divide each term in $12 y = - 9$ by $12$ .

$\frac{12 y}{12} = \frac{- 9}{12}$

Step 4.4.5.2

Simplify the left side.

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Step 4.4.5.2.1

Cancel the common factor of $12$ .

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Step 4.4.5.2.1.1

Cancel the common factor.

$\frac{12 y}{12} = \frac{- 9}{12}$

Step 4.4.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 9}{12}$

Step 4.4.5.3

Simplify the right side.

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Step 4.4.5.3.1

Cancel the common factor of $- 9$ and $12$ .

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Step 4.4.5.3.1.1

Factor $3$ out of $- 9$ .

$y = \frac{3 (- 3)}{12}$

Step 4.4.5.3.1.2

Cancel the common factors.

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Step 4.4.5.3.1.2.1

Factor $3$ out of $12$ .

$y = \frac{3 \cdot - 3}{3 \cdot 4}$

Step 4.4.5.3.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot - 3}{3 \cdot 4}$

Step 4.4.5.3.1.2.3

Rewrite the expression.

$y = \frac{- 3}{4}$

Step 4.4.5.3.2

Move the negative in front of the fraction.

$y = - \frac{3}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{3}{4}$

Decimal Form:

$y = - 0.75$