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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].
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].
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