Solve the Inequality for x 16^x>2^24^38^4

$16^{x} > 2^{2} \cdot 4^{3} \cdot 8^{4}$

Step 1

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

$16^{x} > 2^{2} \cdot {(2^{2})}^{3} \cdot 8^{4}$

Step 2

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

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

$16^{x} > 2^{2} \cdot 2^{2 \cdot 3} \cdot 8^{4}$

Multiply $2$ by $3$ .

$16^{x} > 2^{2} \cdot 2^{6} \cdot 8^{4}$

Step 3

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

$16^{x} > 2^{2 + 6} \cdot 8^{4}$

Step 4

Add $2$ and $6$ .

$16^{x} > 2^{8} \cdot 8^{4}$

Step 5

Rewrite $8$ as $2^{3}$ .

$16^{x} > 2^{8} \cdot {(2^{3})}^{4}$

Step 6

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

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

$16^{x} > 2^{8} \cdot 2^{3 \cdot 4}$

Multiply $3$ by $4$ .

$16^{x} > 2^{8} \cdot 2^{12}$

Step 7

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

$16^{x} > 2^{8 + 12}$

Step 8

Add $8$ and $12$ .

$16^{x} > 2^{20}$

Step 9

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

$2^{4 x} > 2^{20}$

Step 10

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

$4 x > 20$

Step 11

Divide each term in $4 x > 20$ by $4$ and simplify.

Divide each term in $4 x > 20$ by $4$ .

$\frac{4 x}{4} > \frac{20}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} > \frac{20}{4}$

Divide $x$ by $1$ .

$x > \frac{20}{4}$

Simplify the right side.

Divide $20$ by $4$ .

$x > 5$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x > 5$

Interval Notation:

$(5, \infty)$

Step 13

Solve the Inequality for x 16^x>2^2*4^3*8^4

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Solve the Inequality for x 16^x>2^24^38^4