Question

mixed exercises
solve each equation or inequality.

1. $8^{4x + 2} = 64$
2. $5^{x - 6} = 125$
3. $625 \geq 5^{a + 8}$
4. $256^{b + 2} = 4^{2 - 2b}$

Explanation:

Problem 13: \( 8^{4x + 2} = 64 \)

Step 1: Express with same base

Rewrite \( 8 \) as \( 2^3 \) and \( 64 \) as \( 2^6 \). So the equation becomes \( (2^3)^{4x + 2} = 2^6 \).

Step 2: Simplify exponents

Using the power - of - a - power rule \( (a^m)^n=a^{mn} \), we get \( 2^{3(4x + 2)}=2^6 \), which simplifies to \( 2^{12x+6}=2^6 \).

Step 3: Set exponents equal

Since the bases are the same and the equation holds, we can set the exponents equal: \( 12x + 6=6 \).

Step 4: Solve for x

Subtract 6 from both sides: \( 12x=6 - 6=0 \). Then divide both sides by 12: \( x = 0 \).

Step 1: Express with same base

Rewrite \( 125 \) as \( 5^3 \). The equation becomes \( 5^{x - 6}=5^3 \).

Step 2: Set exponents equal

Since the bases are the same, we set the exponents equal: \( x-6 = 3 \).

Step 3: Solve for x

Add 6 to both sides: \( x=3 + 6=9 \).

Step 1: Express with same base

Rewrite \( 625 \) as \( 5^4 \). The inequality becomes \( 5^4\geq5^{a + 8} \).

Step 2: Use exponential function property

For the exponential function \( y = 5^x \), since the base \( 5>1 \), the function is increasing. So if \( 5^4\geq5^{a + 8} \), then \( 4\geq a + 8 \).

Step 3: Solve for a

Subtract 8 from both sides: \( 4-8\geq a \), which simplifies to \( - 4\geq a \) or \( a\leq - 4 \).

Answer:

\( x = 0 \)

Problem 14: \( 5^{x - 6}=125 \)