Let's solve the equation step by step.

The given equation is:

\[
2(2^x + 2^x + 2^x + 2^x) = 2^{20}
\]

### Step 1: Simplify the Left Side

First, notice that the expression inside the parentheses consists of four identical terms, \(2^x\). Therefore, we can rewrite the sum inside the parentheses as:

\[
2^x + 2^x + 2^x + 2^x = 4 \times 2^x
\]

So the equation becomes:

\[
2 \times 4 \times 2^x = 2^{20}
\]

### Step 2: Simplify Further

Next, simplify the left side:

\[
2 \times 4 = 8
\]

Thus, the equation now becomes:

\[
8 \times 2^x = 2^{20}
\]

### Step 3: Express 8 as a Power of 2

We know that \(8\) can be written as \(2^3\). Therefore, we can rewrite the equation as:

\[
2^3 \times 2^x = 2^{20}
\]

### Step 4: Combine the Exponents

According to the properties of exponents, when you multiply like bases, you add the exponents:

\[
2^{x+3} = 2^{20}
\]

### Step 5: Set the Exponents Equal

Since the bases are the same, we can set the exponents equal to each other:

\[
x + 3 = 20
\]

### Step 6: Solve for \(x\)

Now, solve for \(x\):

\[
x = 20 - 3 = 17
\]

### Final Answer

The solution to the equation is:

\[
\boxed{17}
\]

Would you like more details or have any questions?

Here are some related questions:

1. What are the properties of exponents that help solve equations like this?
2. How can you express numbers like 8, 16, or 32 as powers of 2?
3. What are some real-world applications of exponential equations?
4. Can this equation be solved graphically, and if so, how?
5. How would the solution change if the equation were \(2(2^x + 2^x) = 2^{20}\)?

**Tip:** When dealing with exponential equations, always try to express all terms with the same base to simplify the equation.

Learn how to solve the exponential equation 2(2^x + 2^x + 2^x + 2^x) = 2^20 with a detailed step-by-step solution. This problem involves understanding exponents and powers of 2. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation