QuestionFind ' $\mathrm{p}^{\prime}$ ' if $5^{p-3} \times 3^{2 p-8} = 225$ .

Studdy Solution

STEP 1

Assumptions1. We are given the equation $5^{p-3} \times3^{ p-8}=225$ . We need to find the value of $p$
3. We assume that $p$ is a real number

STEP 2

First, we need to simplify the right side of the equation. The number225 can be expressed as the product of powers of5 and.
$225 =5^2 \times^2$

STEP 3

Now, we can rewrite the given equation by replacing225 with $5^2 \times3^2$ .
$5^{p-3} \times3^{2 p-8}=5^2 \times3^2$

STEP 4

Next, we can separate the terms with the same base on both sides of the equation.
$^{p-3} =^2 \quad and \quad3^{2 p-8}=3^2$

STEP 5

From the equation $5^{p-3} =5^2$ , we can equate the exponents because the bases are equal.
$p-3 =2$

STEP 6

olve the equation $p-3 =2$ for $p$ .
$p =2 +3$

STEP 7

Calculate the value of $p$ .
$p =2 +3 =5$

STEP 8

Now, we can substitute $p =5$ into the second equation $3^{2 p-8}=3^2$ to verify if it holds true.
$3^{2 \times5 -8}=3^2$

STEP 9

implify the equation $3^{2 \times5 -8}=3^2$ .
$3^{ -8}=3^2$

STEP 10

Calculate the exponent on the left side of the equation.
$3^{2}=3^2$

STEP 11

Since 3^ =3^ holds true, we can conclude that the value of $p$ is indeed5.
The value of $p$ is5.

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