To solve the equation [tex]\( 5 \sqrt{5} \times 5^3 \div 5^{-3 / 2} = 5^{x + 2} \)[/tex], follow these steps:
1. Convert all terms to the same base:
- The term [tex]\(\sqrt{5}\)[/tex] can be written as [tex]\(5^{1/2}\)[/tex].
- The term [tex]\(5^3\)[/tex] is already in the form of [tex]\(5\)[/tex] raised to a power.
- The term [tex]\(5^{-3/2}\)[/tex] is already in the form of [tex]\(5\)[/tex] raised to a power.
2. Rewrite the expression using the exponents:
[tex]\[
5 \times 5^{1/2} \times 5^3 \div 5^{-3/2}
\][/tex]
3. Combine the exponents:
According to the properties of exponents:
[tex]\[
a^m \times a^n = a^{m+n} \quad \text{and} \quad a^m \div a^n = a^{m-n}
\][/tex]
Combine the exponents as follows:
[tex]\[
5^1 \times 5^{1/2} \times 5^3 \div 5^{-3/2}
\][/tex]
Add the exponents for multiplication and subtract the exponent for division:
[tex]\[
1 + \frac{1}{2} + 3 - \left(-\frac{3}{2}\right)
\][/tex]
4. Simplify the exponents:
Combine the exponents:
[tex]\[
1 + \frac{1}{2} + 3 + \frac{3}{2}
\][/tex]
Simplify step-by-step:
[tex]\[
1 + \frac{1}{2} = 1.5
\][/tex]
[tex]\[
1.5 + 3 = 4.5
\][/tex]
[tex]\[
4.5 + \frac{3}{2} = 6
\][/tex]
So, the combined exponent is [tex]\(6\)[/tex].
5. Compare both sides of the equation:
We have:
[tex]\[
5^{6} = 5^{x + 2}
\][/tex]
6. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
6 = x + 2
\][/tex]
7. Solve for [tex]\(x\)[/tex]:
Subtract 2 from both sides:
[tex]\[
x = 6 - 2
\][/tex]
[tex]\[
x = 4
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\(4\)[/tex].
