QuestionII. Solve each equation. (5 points each with checking) 1.) $3^{n-2}=27$ 3.) $5^{n-3}=\frac{1}{25}$ 2.) $\left(\frac{1}{9}\right)^{m}=81^{m+4}$ 4.) $\left(\frac{1}{7} y-3=243\right.$

Studdy Solution

STEP 1

1. The equations involve exponential expressions.
2. We will need to use properties of exponents to solve for the variables.
3. Checking involves substituting the solution back into the original equation to verify.

STEP 2

1. Solve equation 1: $3^{n-2} = 27$ .
2. Solve equation 3: $5^{n-3} = \frac{1}{25}$ .
3. Solve equation 2: $\left(\frac{1}{9}\right)^{m} = 81^{m+4}$ .
4. Solve equation 4: $\left(\frac{1}{7} y - 3 = 243\right)$ .

STEP 3

For the equation $3^{n-2} = 27$ , express 27 as a power of 3:
$27 = 3^3$
So, the equation becomes:
$3^{n-2} = 3^3$
Since the bases are the same, equate the exponents:
$n - 2 = 3$
Solve for $n$ :
$n = 3 + 2$ $n = 5$
Check: Substitute $n = 5$ back into the original equation:
$3^{5-2} = 3^3 = 27$
The solution is correct.

STEP 4

For the equation $5^{n-3} = \frac{1}{25}$ , express $\frac{1}{25}$ as a power of 5:
$\frac{1}{25} = 5^{-2}$
So, the equation becomes:
$5^{n-3} = 5^{-2}$
Since the bases are the same, equate the exponents:
$n - 3 = -2$
Solve for $n$ :
$n = -2 + 3$ $n = 1$
Check: Substitute $n = 1$ back into the original equation:
$5^{1-3} = 5^{-2} = \frac{1}{25}$
The solution is correct.

STEP 5

For the equation $\left(\frac{1}{9}\right)^{m} = 81^{m+4}$ , express both sides with the same base:
$\frac{1}{9} = 9^{-1} = 3^{-2}$ $81 = 3^4$
So, the equation becomes:
$(3^{-2})^{m} = (3^4)^{m+4}$
Simplify both sides:
$3^{-2m} = 3^{4(m+4)}$
Since the bases are the same, equate the exponents:
$-2m = 4(m+4)$
Expand and solve for $m$ :
$-2m = 4m + 16$ $-2m - 4m = 16$ $-6m = 16$ $m = -\frac{16}{6}$ $m = -\frac{8}{3}$
Check: Substitute $m = -\frac{8}{3}$ back into the original equation and verify both sides are equal.

STEP 6

For the equation $\left(\frac{1}{7} y - 3 = 243\right)$ , solve for $y$ :
First, isolate the term with $y$ :
$\frac{1}{7} y - 3 = 243$
Add 3 to both sides:
$\frac{1}{7} y = 243 + 3$ $\frac{1}{7} y = 246$
Multiply both sides by 7 to solve for $y$ :
$y = 246 \times 7$ $y = 1722$
Check: Substitute $y = 1722$ back into the original equation:
$\frac{1}{7} \times 1722 - 3 = 243$
Calculate:
$246 - 3 = 243$
The solution is correct.
The solutions are:
1. $n = 5$
2. $n = 1$
3. $m = -\frac{8}{3}$
4. $y = 1722$

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