7. Simplify the expression $$ \frac{3 a^{4} imes(10 a)^{3}}{\left(5 a^{2} ight)^{3}} $$

Question

7. Simplify the expression $$ \frac{3 a^{4} 	imes(10 a)^{3}}{\left(5 a^{2}ight)^{3}} $$

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{3 a^{4} \times (10 a)^{3}}{\left(5 a^{2}\right)^{3}}
$$

follow these steps:

### 1. Simplify the Numerator and the Denominator Separately

**Numerator:**
$$
3 a^{4} \times (10 a)^{3}
$$

First, expand $$(10 a)^{3}$$:
$$
(10 a)^{3} = 10^{3} \times a^{3} = 1000 \times a^{3}
$$

Now, multiply by $$3 a^{4}$$:
$$
3 a^{4} \times 1000 a^{3} = 3000 \times a^{4+3} = 3000 a^{7}
$$

**Denominator:**
$$
\left(5 a^{2}\right)^{3}
$$

Expand the denominator:
$$
\left(5 a^{2}\right)^{3} = 5^{3} \times \left(a^{2}\right)^{3} = 125 \times a^{6} = 125 a^{6}
$$

### 2. Combine the Simplified Numerator and Denominator

Now, the expression is:
$$
\frac{3000 a^{7}}{125 a^{6}}
$$

### 3. Simplify the Coefficients and the Variables Separately

**Coefficients:**
$$
\frac{3000}{125} = 24
$$

**Variables:**
$$
\frac{a^{7}}{a^{6}} = a^{7-6} = a^{1} = a
$$

### 4. Write the Final Simplified Expression

Combining the simplified coefficient and variable:
$$
24 a
$$

**Answer:** $$24\,a$$
The simplified expression is $$24a$$.

7. Simplify the expression \( \frac{3 a^{4} \times(10 a)^{3}}{\left(5 a^{2}\right)^{3}} \)