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Question 2

If [tex]\((25)^{7.5} \times (5)^{2.5} \div (125)^{1.5} = 5^x\)[/tex],

What is the value of [tex]\(x\)[/tex]?

A. 12
B. 17
C. 19
D. 13


Sagot :

Let's solve the given mathematical expression step-by-step:

[tex]\[ (25)^{7.5} \times (5)^{2.5} \div (125)^{1.5} = 5^x \][/tex]

First, we need to express all terms with the same base. We know that:

[tex]\[ 25 = 5^2 \quad \text{and} \quad 125 = 5^3 \][/tex]

So, we rewrite the equation using these equivalent expressions:

[tex]\[ (5^2)^{7.5} \times (5)^{2.5} \div (5^3)^{1.5} \][/tex]

Next, we use the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to simplify the exponents:

[tex]\[ 5^{2 \cdot 7.5} \times 5^{2.5} \div 5^{3 \cdot 1.5} \][/tex]

Calculate the exponents:

[tex]\[ 5^{15} \times 5^{2.5} \div 5^{4.5} \][/tex]

Now, we combine the exponents of the terms with the same base. Recall the rule [tex]\(a^m \times a^n = a^{m+n}\)[/tex] and [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:

[tex]\[ 5^{15 + 2.5} \div 5^{4.5} \][/tex]

Simplify the exponents:

[tex]\[ 5^{17.5} \div 5^{4.5} \][/tex]

Now, subtract the exponents in the denominator from the exponent in the numerator:

[tex]\[ 5^{17.5 - 4.5} \][/tex]

Perform the subtraction:

[tex]\[ 5^{13} \][/tex]

Therefore, we have:

[tex]\[ 5^x = 5^{13} \][/tex]

This implies that:

[tex]\[ x = 13 \][/tex]

So, the value of [tex]\(x\)[/tex] is:

[tex]\[ \boxed{13} \][/tex]