7. (a) $$ 3^{x}=\sqrt[4]{3^{5}} $$ Find the value of $$ x $$. $$ \square $$ :mwilecth givinu $$ \square $$ -hubloin bluedz uoy (b) Simplify $$ \left(32 y^{15} ight)^{\frac{2}{2}} $$. 8. Find the values of $$ m $$ and $$ n $$. (a) $$ 2^{m}=0.125 $$ Anstiots ad -104. fais inuanm pa patini 6 al nothamolal gmtos insingho iil gized smae griad tamal bidssingosen -ationh mionalias 10 - inje smoz Aifive noitacmetm litahupe sit (b) $$ 2^{4 n} imes 2^{2 n}=512 $$

Question

7. (a) $$ 3^{x}=\sqrt[4]{3^{5}} $$ Find the value of $$ x $$. $$ \square $$ :mwilecth givinu $$ \square $$ -hubloin bluedz uoy (b) Simplify $$ \left(32 y^{15}ight)^{\frac{2}{2}} $$. 8. Find the values of $$ m $$ and $$ n $$. (a) $$ 2^{m}=0.125 $$ Anstiots ad -104. fais inuanm pa patini 6 al nothamolal gmtos insingho iil gized smae griad tamal bidssingosen -ationh mionalias 10 - inje smoz Aifive noitacmetm litahupe sit (b) $$ 2^{4 n} 	imes 2^{2 n}=512 $$

UpStudy AI · Accepted Answer

Solve the equation by following steps:
- step0: Solve for $$m$$:
  $$2^{m}=0.125$$
- step1: Rewrite the expression:
  $$2^{m}=\frac{1}{8}$$
- step2: Rewrite the expression:
  $$2^{m}=2^{-3}$$
- step3: Set the exponents equal:
  $$m=-3$$
Calculate or simplify the expression $$ (32 y^{15})^{2/2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(32y^{15}\right)^{\frac{2}{2}}$$
- step1: Divide the terms:
  $$\left(32y^{15}\right)^{1}$$
- step2: Calculate:
  $$32y^{15}$$
Solve the equation $$ 2^{4n} \times 2^{2n}=512 $$.
Solve the equation by following steps:
- step0: Solve for $$n$$:
  $$2^{4n}\times 2^{2n}=512$$
- step1: Multiply the terms:
  $$2^{6n}=512$$
- step2: Rewrite in exponential form:
  $$2^{6n}=2^{9}$$
- step3: Set the exponents equal:
  $$6n=9$$
- step4: Divide both sides:
  $$\frac{6n}{6}=\frac{9}{6}$$
- step5: Divide the numbers:
  $$n=\frac{3}{2}$$
Solve the equation $$ 3^{x}=3^{5/4} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$3^{x}=3^{\frac{5}{4}}$$
- step1: Set the exponents equal:
  $$x=\frac{5}{4}$$
Let's solve the problems step by step.

### Problem 7

#### (a) Solve $$ 3^{x}=\sqrt[4]{3^{5}} $$

To solve this equation, we can rewrite the right side:

$$
\sqrt[4]{3^{5}} = 3^{5/4}
$$

Now we have:

$$
3^{x} = 3^{5/4}
$$

Since the bases are the same, we can set the exponents equal to each other:

$$
x = \frac{5}{4}
$$

Thus, the value of $$ x $$ is:

$$
x = 1.25
$$

#### (b) Simplify $$ \left(32 y^{15}\right)^{\frac{2}{2}} $$

The expression simplifies as follows:

$$
\left(32 y^{15}\right)^{\frac{2}{2}} = 32 y^{15}
$$

So, the simplified form is:

$$
32 y^{15}
$$

---

### Problem 8

#### (a) Solve $$ 2^{m}=0.125 $$

We can express $$ 0.125 $$ as a power of $$ 2 $$:

$$
0.125 = \frac{1}{8} = 2^{-3}
$$

Thus, we have:

$$
2^{m} = 2^{-3}
$$

Setting the exponents equal gives us:

$$
m = -3
$$

#### (b) Solve $$ 2^{4n} \times 2^{2n}=512 $$

We can combine the left side:

$$
2^{4n + 2n} = 2^{6n}
$$

Now, we know that $$ 512 = 2^9 $$, so we can set the exponents equal:

$$
6n = 9
$$

Solving for $$ n $$:

$$
n = \frac{9}{6} = \frac{3}{2} = 1.5
$$

### Summary of Results
- For $$ 7(a) $$, $$ x = 1.25 $$
- For $$ 7(b) $$, the simplified expression is $$ 32 y^{15} $$
- For $$ 8(a) $$, $$ m = -3 $$
- For $$ 8(b) $$, $$ n = 1.5 $$
- For problem 7(a), $$ x = 1.25 $$. - For problem 7(b), the simplified expression is $$ 32 y^{15} $$. - For problem 8(a), $$ m = -3 $$. - For problem 8(b), $$ n = 1.5 $$.

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