Basic Math Examples

\sqrt[3]{\sqrt{2^{0.5 m}}} = 4

$\sqrt[3]{\sqrt{2^{0.5 m}}} = 4$

Step 1

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

{\sqrt[3]{\sqrt{2^{0.5 m}}}}^{3} = 4^{3}

${\sqrt[3]{\sqrt{2^{0.5 m}}}}^{3} = 4^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt[3]{\sqrt{2^{0.5 m}}}

$\sqrt[3]{\sqrt{2^{0.5 m}}}$ as

{\sqrt{2^{0.5 m}}}^{\frac{1}{3}}

${\sqrt{2^{0.5 m}}}^{\frac{1}{3}}$ .

{({\sqrt{2^{0.5 m}}}^{\frac{1}{3}})}^{3} = 4^{3}

${({\sqrt{2^{0.5 m}}}^{\frac{1}{3}})}^{3} = 4^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify

{({\sqrt{2^{0.5 m}}}^{\frac{1}{3}})}^{3}

${({\sqrt{2^{0.5 m}}}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in

{({\sqrt{2^{0.5 m}}}^{\frac{1}{3}})}^{3}

${({\sqrt{2^{0.5 m}}}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

{\sqrt{2^{0.5 m}}}^{\frac{1}{3} \cdot 3} = 4^{3}

${\sqrt{2^{0.5 m}}}^{\frac{1}{3} \cdot 3} = 4^{3}$

Step 2.2.1.1.2

Cancel the common factor of

3

$3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${\sqrt{2^{0.5 m}}}^{\frac{1}{3} \cdot 3} = 4^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${\sqrt{2^{0.5 m}}}^{1} = 4^{3}$

Step 2.2.1.2

Simplify.

$\sqrt{2^{0.5 m}} = 4^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise

$4$ to the power of

$3$ .

$\sqrt{2^{0.5 m}} = 64$

Step 3

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

${\sqrt{2^{0.5 m}}}^{2} = 64^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2^{0.5 m}}$ as

$2^{\frac{0.5 m}{2}}$ .

${(2^{\frac{0.5 m}{2}})}^{2} = 64^{2}$

Step 4.2

Factor

$0.5$ out of

$0.5 m$ .

${(2^{\frac{0.5 (m)}{2}})}^{2} = 64^{2}$

Step 4.3

Factor

$2$ out of

$2$ .

${(2^{\frac{0.5 (m)}{2 (1)}})}^{2} = 64^{2}$

Step 4.4

Separate fractions.

${(2^{\frac{0.5}{2} \cdot \frac{m}{1}})}^{2} = 64^{2}$

Step 4.5

Divide

$0.5$ by

$2$ .

${(2^{0.25 \frac{m}{1}})}^{2} = 64^{2}$

Step 4.6

Cancel the common factor of

$0.25$ .

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Step 4.6.1

Factor

$0.25$ out of

$1$ .

${(2^{0.25 \frac{m}{0.25 \cdot 4}})}^{2} = 64^{2}$

Step 4.6.2

Cancel the common factor.

${(2^{0.25 \frac{m}{0.25 \cdot 4}})}^{2} = 64^{2}$

Step 4.6.3

Rewrite the expression.

${(2^{\frac{m}{4}})}^{2} = 64^{2}$

Step 4.7

Simplify the left side.

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Step 4.7.1

Multiply the exponents in

${(2^{\frac{m}{4}})}^{2}$ .

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Step 4.7.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$2^{\frac{m}{4} \cdot 2} = 64^{2}$

Step 4.7.1.2

Cancel the common factor of

$2$ .

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Step 4.7.1.2.1

Factor

$2$ out of

$4$ .

$2^{\frac{m}{2 (2)} \cdot 2} = 64^{2}$

Step 4.7.1.2.2

Cancel the common factor.

$2^{\frac{m}{2 \cdot 2} \cdot 2} = 64^{2}$

Step 4.7.1.2.3

Rewrite the expression.

$2^{\frac{m}{2}} = 64^{2}$

Step 4.8

Simplify the right side.

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Step 4.8.1

Raise

$64$ to the power of

$2$ .

$2^{\frac{m}{2}} = 4096$

Step 5

Solve for

$m$ .

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Step 5.1

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

$2^{\frac{m}{2}} = 2^{12}$

Step 5.2

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

$\frac{m}{2} = 12$

Step 5.3

Solve for

$m$ .

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Step 5.3.1

Multiply both sides of the equation by

$2$ .

$2 \frac{m}{2} = 2 \cdot 12$

Step 5.3.2

Simplify both sides of the equation.

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Step 5.3.2.1

Simplify the left side.

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Step 5.3.2.1.1

Cancel the common factor of

$2$ .

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Step 5.3.2.1.1.1

Cancel the common factor.

$2 \frac{m}{2} = 2 \cdot 12$

Step 5.3.2.1.1.2

Rewrite the expression.

$m = 2 \cdot 12$

Step 5.3.2.2

Simplify the right side.

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Step 5.3.2.2.1

Multiply

$2$ by

$12$ .

$m = 24$