Algebra Examples

$\frac{512^{x - 2}}{{(\frac{1}{64})}^{3 x}} = 512$

Step 1

Multiply both sides by ${(\frac{1}{64})}^{3 x}$ .

$\frac{512^{x - 2}}{{(\frac{1}{64})}^{3 x}} {(\frac{1}{64})}^{3 x} = 512 {(\frac{1}{64})}^{3 x}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Cancel the common factor of ${(\frac{1}{64})}^{3 x}$ .

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

$\frac{512^{x - 2}}{{(\frac{1}{64})}^{3 x}} {(\frac{1}{64})}^{3 x} = 512 {(\frac{1}{64})}^{3 x}$

Step 2.1.1.2

Rewrite the expression.

$512^{x - 2} = 512 {(\frac{1}{64})}^{3 x}$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Simplify $512 {(\frac{1}{64})}^{3 x}$ .

Tap for more steps...

Step 2.2.1.1

Simplify the expression.

Tap for more steps...

Step 2.2.1.1.1

Apply the product rule to $\frac{1}{64}$ .

$512^{x - 2} = 512 \frac{1^{3 x}}{64^{3 x}}$

Step 2.2.1.1.2

One to any power is one.

$512^{x - 2} = 512 \frac{1}{64^{3 x}}$

Step 2.2.1.2

Combine $512$ and $\frac{1}{64^{3 x}}$ .

$512^{x - 2} = \frac{512}{64^{3 x}}$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Move $64^{3 x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$512^{x - 2} = 512 \cdot 64^{- 3 x}$

Step 3.2

Rewrite $512$ as $2^{9}$ .

$512^{x - 2} = 2^{9} \cdot 64^{- 3 x}$

Step 3.3

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

$512^{x - 2} = 2^{9} \cdot {(2^{6})}^{- 3 x}$

Step 3.4

Multiply the exponents in ${(2^{6})}^{- 3 x}$ .

Tap for more steps...

Step 3.4.1

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

$512^{x - 2} = 2^{9} \cdot 2^{6 (- 3 x)}$

Step 3.4.2

Multiply $- 3$ by $6$ .

$512^{x - 2} = 2^{9} \cdot 2^{- 18 x}$

Step 3.5

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

$512^{x - 2} = 2^{9 - 18 x}$

Step 3.6

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

$2^{9 (x - 2)} = 2^{9 - 18 x}$

Step 3.7

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

$9 (x - 2) = 9 - 18 x$

Step 3.8

Solve for $x$ .

Tap for more steps...

Step 3.8.1

Simplify $9 (x - 2)$ .

Tap for more steps...

Step 3.8.1.1

Rewrite.

$0 + 0 + 9 (x - 2) = 9 - 18 x$

Step 3.8.1.2

Simplify by adding zeros.

$9 (x - 2) = 9 - 18 x$

Step 3.8.1.3

Apply the distributive property.

$9 x + 9 \cdot - 2 = 9 - 18 x$

Step 3.8.1.4

Multiply $9$ by $- 2$ .

$9 x - 18 = 9 - 18 x$

Step 3.8.2

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

Tap for more steps...

Step 3.8.2.1

Add $18 x$ to both sides of the equation.

$9 x - 18 + 18 x = 9$

Step 3.8.2.2

Add $9 x$ and $18 x$ .

$27 x - 18 = 9$

Step 3.8.3

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

Tap for more steps...

Step 3.8.3.1

Add $18$ to both sides of the equation.

$27 x = 9 + 18$

Step 3.8.3.2

Add $9$ and $18$ .

$27 x = 27$

Step 3.8.4

Divide each term in $27 x = 27$ by $27$ and simplify.

Tap for more steps...

Step 3.8.4.1

Divide each term in $27 x = 27$ by $27$ .

$\frac{27 x}{27} = \frac{27}{27}$

Step 3.8.4.2

Simplify the left side.

Tap for more steps...

Step 3.8.4.2.1

Cancel the common factor of $27$ .

Tap for more steps...

Step 3.8.4.2.1.1

Cancel the common factor.

$\frac{27 x}{27} = \frac{27}{27}$

Step 3.8.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{27}{27}$

Step 3.8.4.3

Simplify the right side.

Tap for more steps...

Step 3.8.4.3.1

Divide $27$ by $27$ .

$x = 1$