Question

Determine the value of the variable in the product of powers.

What is the value of x in the product of powers 55 · 5x = 52 ?
x =
What is the value of y in the product of p-3 · p-6 = py
y =
What is the value of n in th product of 92 · 9-7 · 9n = 9-5 = StartFraction 1 Over 9 Superscript 5 EndFraction?
n =

Answer 1

To solve each of these equations, we can use the product of powers property, which states that \( a^m \cdot a^n = a^{m+n} \).

**1. Solve for x in the equation:**
\[ 55 \cdot 5^x = 5^2 \]

First, express \( 55 \) in terms of base \( 5 \):
\[ 5^2 = 25 \]

So,
\[ 5^2 \cdot 5^x = 5^2 \]

Now we can set the bases equal:
\[ 5^{2+x} = 5^2 \]

Since the bases are the same, we can set the exponents equal:
\[ 2+x = 2 \]

Now, solve for \( x \):
\[ x = 2 - 2 \]
\[ x = 0 \]

**Therefore,** \( x = 0 \).

---

**2. Solve for y in the equation:**
\[ p^{-3} \cdot p^{-6} = p^y \]

Using the product of powers, combine the left side:
\[ p^{-3 + (-6)} = p^y \]
\[ p^{-9} = p^y \]

Set the exponents equal:
\[ -9 = y \]

**Therefore,** \( y = -9 \).

---

**3. Solve for n in the equation:**
\[ 9^2 \cdot 9^{-7} \cdot 9^n = 9^{-5} \]

Combine the left side:
\[ 9^{2 + (-7) + n} = 9^{-5} \]
\[ 9^{-5 + n} = 9^{-5} \]

Set the exponents equal:
\[ -5 + n = -5 \]

Solve for \( n \):
\[ n = -5 + 5 \]
\[ n = 0 \]

**Therefore,** \( n = 0 \).

---

To summarize:
- \( x = 0 \)
- \( y = -9 \)
- \( n = 0 \)