Find the value of x if $[ 1 ÷ 64^{(x + 2)}]= [4^{(x − 3)} ÷ 16^x ]$

1. Analyze the question

We are given an exponential equation:

Our goal is to find the value of $x$ by expressing all terms with a common base.

2. Explain the concept

To solve exponential equations, we use the Law of Indices. The most effective strategy is to express all numbers as powers of the same base. In this case, notice that $4, 16,$ and $64$ are all powers of $4$ (or $2$). Using base $4$ is the simplest approach:

• $4 = 4^1$
• $16 = 4^2$
• $64 = 4^3$
• Recall that $\frac{1}{a^n} = a^{-n}$

3. Show the solution

Step 1: Rewrite all terms using base 4.

• Left side: $\frac{1}{64^{(x + 2)}} = \frac{1}{(4^3)^{(x + 2)}} = \frac{1}{4^{3x + 6}} = 4^{-(3x + 6)}$
• Right side: $\frac{4^{(x - 3)}}{16^x} = \frac{4^{(x - 3)}}{(4^2)^x} = \frac{4^{(x - 3)}}{4^{2x}}$

Step 2: Simplify the right side using the division law ($a^m \div a^n = a^{m-n}$).

• Right side: $4^{(x - 3) - 2x} = 4^{-x - 3}$

Step 3: Equate the exponents. Now we have:

Since the bases are the same, we can equate the exponents:

Step 4: Solve for $x$.

Add $3x$ to both sides:

Add $3$ to both sides:

Divide by $2$:

4. Identify the answer

Comparing our result to the given options, we find that the correct choice is D.

5. Tip

When dealing with indices, always look for the smallest common base. If you see $4, 8, 16, 32, 64$, try converting everything to base $2$. It makes the arithmetic much simpler and reduces the chance of making a sign error!