How to simplify Raising One Power to Another in Algebra

Hello! Today, we are going to learn about an important concept in algebra called “raising one power to another.” This might sound a bit complicated, but I’ll break it down into simple steps to make it easy to understand.

What Are Exponents?

Before we dive into the main topic, let’s quickly review what exponents are. An exponent tells us how many times to multiply a base by itself.

For example, in 2³ 

• 2 is base and 3 is exponent
• 2³ means 2 × 2 × 2 = 8

Raising a Power to Another Power

Now, let’s talk about raising a power to another power. This means we have a base with an exponent, and we raise that whole expression to another exponent.

For example:

• (2³)² 

Simplifying (2³)²

Let’s break down the steps to simplify this expression.

1. Understand the Expression:
   • (2³)² means we have 2³, and we are raising it to the power of 2.
2. Use the Rule for Powers of Powers:
   • When you raise a power to another power, you multiply the exponents. The general rule is (a^m)ⁿ = a^m×n
3. Apply the Rule:
   • In our example, a = 2, m = 3 and n = 2.
   • So, (2³)²  = 2^3×2 ⇒ 2⁶ .
4. Simplify the Result:
   • Now, calculate 2⁶:
   • 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 ⇒ 64.

So, (2³)² = 64

More Examples

Let’s try a few more examples to make sure we understand this concept.

Example 1: (3²)³

1. Understand the Expression:
   • (3²)³ means we have 3², and we are raising it to the power of 3.
2. Apply the Rule:
   • a = 3, m = 2 and n = 3.
   • (3²)³ =  3^2×3 ⇒ 3⁶.
3. Simplify the Result:
   • Now, calculate 3⁶:
   • 3⁶ = 3 × 3 × 3 × 3 × 3 × 3 = 729 .

So, (3²)³ = 729 .

Example 2: (5⁴)²

1. Understand the Expression:
   • (5⁴)²  means we have 5⁴, and we are raising it to the power of 2.
2. Apply the Rule:
   • a = 5, m = 4 and n = 2.
   • (5⁴)²  = 5^4×2  ⇒ 5⁸.
3. Simplify the Result:
   • Now, calculate 5⁸:
   • This one is a bit trickier to calculate by hand, but it’s still possible.
   • 5⁸ = 5×5×5×5×5×5×5×5 ⇒.390,625

So, (5⁴)² = 390,625.

Common Mistakes to Avoid

1. Forgetting to Multiply the Exponents:
   • Remember, when you raise a power to another power, you multiply the exponents. Don’t add them!
2. Not Simplifying the Base Expression First:
   • Make sure you understand the base expression before applying the rule. If the base itself is a complex expression, simplify it first.
3. Misinterpreting the Rule:
   • The rule (a^m)ⁿ = a^m×n is specific to raising a power to another power. Don’t confuse it with other exponent rules.

Practice Problems

Now it’s your turn to practice! Try simplifying these expressions on your own:

1. (4²)³
2. (6³)²
3. (7²)⁴
4. (2⁵)³

Remember to follow the steps we discussed. Check your answers below to see how you did.

Answers to Practice Problems

1. (4²)³ = 4096
2. (6³)² = 46,656
3. (7²)⁴ = 5,764,801
4. (2⁵)³ = 32,768

Great job! Keep practicing, and you’ll master the concept of raising one power to another in no time. Happy learning!