Basic Math Examples

$\frac{125 \cdot 25 \cdot {(5^{\frac{1}{4}})}^{3}}{\sqrt[3]{5}} = 5^{p}$

Step 1

Rewrite the equation as $5^{p} = \frac{125 \cdot 25 \cdot {(5^{\frac{1}{4}})}^{3}}{\sqrt[3]{5}}$ .

$5^{p} = \frac{125 \cdot 25 \cdot {(5^{\frac{1}{4}})}^{3}}{\sqrt[3]{5}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$5^{p} = \frac{125 \cdot 25 \cdot {(5^{\frac{1}{4}})}^{3}}{5^{\frac{1}{3}}}$

Step 3

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

$5^{p} = \frac{125 \cdot 25 \cdot 5^{\frac{1}{4} \cdot 3}}{5^{\frac{1}{3}}}$

Step 4

Move $5^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$5^{p} = 125 \cdot 25 \cdot 5^{\frac{1}{4} \cdot 3} \cdot 5^{- \frac{1}{3}}$

Step 5

Multiply $125$ by $25$ .

$5^{p} = 3125 \cdot 5^{\frac{1}{4} \cdot 3} \cdot 5^{- \frac{1}{3}}$

Step 6

Rewrite $3125$ as $5^{5}$ .

$5^{p} = 5^{5} \cdot 5^{\frac{1}{4} \cdot 3} \cdot 5^{- \frac{1}{3}}$

Step 7

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

$5^{p} = 5^{5 + \frac{1}{4} \cdot 3} \cdot 5^{- \frac{1}{3}}$

Step 8

Multiply $5^{5 + \frac{1}{4} \cdot 3}$ by $5^{- \frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 8.1

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

$5^{p} = 5^{5 + \frac{1}{4} \cdot 3 - \frac{1}{3}}$

Step 8.2

Combine $\frac{1}{4}$ and $3$ .

$5^{p} = 5^{5 + \frac{3}{4} - \frac{1}{3}}$

Step 8.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$5^{p} = 5^{5 \cdot \frac{4}{4} + \frac{3}{4} - \frac{1}{3}}$

Step 8.4

Combine $5$ and $\frac{4}{4}$ .

$5^{p} = 5^{\frac{5 \cdot 4}{4} + \frac{3}{4} - \frac{1}{3}}$

Step 8.5

Combine the numerators over the common denominator.

$5^{p} = 5^{\frac{5 \cdot 4 + 3}{4} - \frac{1}{3}}$

Step 8.6

Simplify the numerator.

Tap for more steps...

Step 8.6.1

Multiply $5$ by $4$ .

$5^{p} = 5^{\frac{20 + 3}{4} - \frac{1}{3}}$

Step 8.6.2

Add $20$ and $3$ .

$5^{p} = 5^{\frac{23}{4} - \frac{1}{3}}$

Step 8.7

To write $\frac{23}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5^{p} = 5^{\frac{23}{4} \cdot \frac{3}{3} - \frac{1}{3}}$

Step 8.8

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$5^{p} = 5^{\frac{23}{4} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{4}{4}}$

Step 8.9

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 8.9.1

Multiply $\frac{23}{4}$ by $\frac{3}{3}$ .

$5^{p} = 5^{\frac{23 \cdot 3}{4 \cdot 3} - \frac{1}{3} \cdot \frac{4}{4}}$

Step 8.9.2

Multiply $4$ by $3$ .

$5^{p} = 5^{\frac{23 \cdot 3}{12} - \frac{1}{3} \cdot \frac{4}{4}}$

Step 8.9.3

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$5^{p} = 5^{\frac{23 \cdot 3}{12} - \frac{4}{3 \cdot 4}}$

Step 8.9.4

Multiply $3$ by $4$ .

$5^{p} = 5^{\frac{23 \cdot 3}{12} - \frac{4}{12}}$

Step 8.10

Combine the numerators over the common denominator.

$5^{p} = 5^{\frac{23 \cdot 3 - 4}{12}}$

Step 8.11

Simplify the numerator.

Tap for more steps...

Step 8.11.1

Multiply $23$ by $3$ .

$5^{p} = 5^{\frac{69 - 4}{12}}$

Step 8.11.2

Subtract $4$ from $69$ .

$5^{p} = 5^{\frac{65}{12}}$

Step 9

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

$p = \frac{65}{12}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$p = \frac{65}{12}$

Decimal Form:

$p = 5.41\overline{6}$

Mixed Number Form:

$p = 5 \frac{5}{12}$