Basic Math Examples

$16^{- 32} - 64^{- 32 - 3}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{16^{32}} - 64^{- 32 - 3}$

Step 1.2

Subtract $3$ from $- 32$ .

$\frac{1}{16^{32}} - 64^{- 35}$

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{16^{32}} - \frac{1}{64^{35}}$

Step 2

To write $\frac{1}{16^{32}}$ as a fraction with a common denominator, multiply by $\frac{64^{35}}{64^{35}}$ .

$\frac{1}{16^{32}} \cdot \frac{64^{35}}{64^{35}} - \frac{1}{64^{35}}$

Step 3

To write $- \frac{1}{64^{35}}$ as a fraction with a common denominator, multiply by $\frac{16^{32}}{16^{32}}$ .

$\frac{1}{16^{32}} \cdot \frac{64^{35}}{64^{35}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4

Write each expression with a common denominator of $16^{32} \cdot 64^{35}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{1}{16^{32}}$ by $\frac{64^{35}}{64^{35}}$ .

$\frac{64^{35}}{16^{32} \cdot 64^{35}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4.2

Rewrite $16$ as $2^{4}$ .

$\frac{64^{35}}{{(2^{4})}^{32} \cdot 64^{35}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4.3

Multiply the exponents in ${(2^{4})}^{32}$ .

Tap for more steps...

Step 4.3.1

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

$\frac{64^{35}}{2^{4 \cdot 32} \cdot 64^{35}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4.3.2

Multiply $4$ by $32$ .

$\frac{64^{35}}{2^{128} \cdot 64^{35}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4.4

Rewrite $64$ as $2^{6}$ .

$\frac{64^{35}}{2^{128} \cdot {(2^{6})}^{35}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4.5

Multiply the exponents in ${(2^{6})}^{35}$ .

Tap for more steps...

Step 4.5.1

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

$\frac{64^{35}}{2^{128} \cdot 2^{6 \cdot 35}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4.5.2

Multiply $6$ by $35$ .

$\frac{64^{35}}{2^{128} \cdot 2^{210}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4.6

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

$\frac{64^{35}}{2^{128 + 210}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4.7

Add $128$ and $210$ .

$\frac{64^{35}}{2^{338}} - \frac{1}{64^{35}} \cdot \frac{16^{32}}{16^{32}}$

Step 4.8

Multiply $\frac{1}{64^{35}}$ by $\frac{16^{32}}{16^{32}}$ .

$\frac{64^{35}}{2^{338}} - \frac{16^{32}}{64^{35} \cdot 16^{32}}$

Step 4.9

Rewrite $64$ as $2^{6}$ .

$\frac{64^{35}}{2^{338}} - \frac{16^{32}}{{(2^{6})}^{35} \cdot 16^{32}}$

Step 4.10

Multiply the exponents in ${(2^{6})}^{35}$ .

Tap for more steps...

Step 4.10.1

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

$\frac{64^{35}}{2^{338}} - \frac{16^{32}}{2^{6 \cdot 35} \cdot 16^{32}}$

Step 4.10.2

Multiply $6$ by $35$ .

$\frac{64^{35}}{2^{338}} - \frac{16^{32}}{2^{210} \cdot 16^{32}}$

Step 4.11

Rewrite $16$ as $2^{4}$ .

$\frac{64^{35}}{2^{338}} - \frac{16^{32}}{2^{210} \cdot {(2^{4})}^{32}}$

Step 4.12

Multiply the exponents in ${(2^{4})}^{32}$ .

Tap for more steps...

Step 4.12.1

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

$\frac{64^{35}}{2^{338}} - \frac{16^{32}}{2^{210} \cdot 2^{4 \cdot 32}}$

Step 4.12.2

Multiply $4$ by $32$ .

$\frac{64^{35}}{2^{338}} - \frac{16^{32}}{2^{210} \cdot 2^{128}}$

Step 4.13

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

$\frac{64^{35}}{2^{338}} - \frac{16^{32}}{2^{210 + 128}}$

Step 4.14

Add $210$ and $128$ .

$\frac{64^{35}}{2^{338}} - \frac{16^{32}}{2^{338}}$

Step 5

Combine the numerators over the common denominator.

$\frac{64^{35} - 16^{32}}{2^{338}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{64^{35} - 16^{32}}{2^{338}}$

Decimal Form:

$2.93873587 \cdot 10^{- 39}$