Question

solve the given equation.
\\(\left(4^{\frac{x}{4}}\
ight)\left(16^{\frac{x}{2}}\
ight)=4^7\\)

\\(x = \square\\) (type an integer or a simplified fraction.)

Explanation:

Step1: Rewrite 16 as power of 4

$16 = 4^2$, so $16^{\frac{x}{2}} = (4^2)^{\frac{x}{2}}$

Step2: Simplify the exponent

Using $(a^m)^n = a^{mn}$, we get $(4^2)^{\frac{x}{2}} = 4^{2 \times \frac{x}{2}} = 4^x$

Step3: Rewrite original equation

The equation becomes $4^{\frac{x}{4}} \times 4^x = 4^7$

Step4: Combine left-hand side terms

Using $a^m \times a^n = a^{m+n}$, we get $4^{\frac{x}{4} + x} = 4^7$

Step5: Equate the exponents

Since bases are equal, $\frac{x}{4} + x = 7$

Step6: Combine like terms

$\frac{x}{4} + \frac{4x}{4} = 7$ → $\frac{5x}{4} = 7$

Step7: Solve for x

Multiply both sides by $\frac{4}{5}$: $x = 7 \times \frac{4}{5}$

Answer:

$\frac{28}{5}$