Simplify the Exponential Expression: 5^(2x) × 5^x

Q: Why do we add exponents when multiplying same bases?

Think of it this way: $ 5^{2x} \times 5^x $ means you're multiplying 5 by itself 2x times, then multiplying by 5 another x times. That's a total of 2x + x times!

Q: What if the bases are different, like 5² × 3²?

The rule only works with same bases! For different bases like $ 5^2 \times 3^2 $, you must calculate each part separately: 25 × 9 = 225. You can't combine different bases.

Q: Can I simplify 2x + x to get the final answer?

Yes! $ 2x + x = 3x $, so $ 5^{2x+x} = 5^{3x} $. But the question asks for $ 5^{2x+x} $ format, so that's the correct choice as written.

Q: How can I remember this rule?

Use the phrase: "Same base, add the race!" The "race" refers to the exponents. When bases match, the exponents get added together in their "race" to the top.

Q: What happens if I have division instead of multiplication?

For division with same bases, you subtract exponents: $ \frac{5^{2x}}{5^x} = 5^{2x-x} = 5^x $. Remember: multiplication = add, division = subtract!

Exponent Rules with Same Base Multiplication

Reduce the following equation:

$5^{2x}\times5^x=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, the multiplication of powers with equal bases (A)

00:11 equals the same base raised to the sum of the exponents (N+M)

00:15 Let's apply this formula to our exercise

00:19 We'll maintain the base and add together the exponents

00:23 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$5^{2x}\times5^x=$

Step-by-step solution

To reduce the expression $5^{2x} \times 5^x$ , we will use the exponent multiplication rule:

When multiplying powers with the same base, add the exponents:
Thus, $5^{2x} \times 5^x = 5^{2x + x}$ .

Hence, the correct choice is: $5^{2x + x}$ .

Final Answer

$5^{2x+x}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: $5^{2x} \times 5^x = 5^{2x+x} = 5^{3x}$
Check: Test with simple values: $5^2 \times 5^1 = 25 \times 5 = 125 = 5^3$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply exponents like $5^{2x \times x} = 5^{2x^2}$ when bases are the same! This gives completely wrong results. Always add exponents when multiplying same bases: $5^{2x} \times 5^x = 5^{2x+x}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add exponents when multiplying same bases?

Think of it this way: $5^{2x} \times 5^x$ means you're multiplying 5 by itself 2x times, then multiplying by 5 another x times. That's a total of 2x + x times!

What if the bases are different, like 5² × 3²?

The rule only works with same bases! For different bases like $5^2 \times 3^2$ , you must calculate each part separately: 25 × 9 = 225. You can't combine different bases.

Can I simplify 2x + x to get the final answer?

Yes! $2x + x = 3x$ , so $5^{2x+x} = 5^{3x}$ . But the question asks for $5^{2x+x}$ format, so that's the correct choice as written.

How can I remember this rule?

Use the phrase: "Same base, add the race!" The "race" refers to the exponents. When bases match, the exponents get added together in their "race" to the top.

What happens if I have division instead of multiplication?

For division with same bases, you subtract exponents: $\frac{5^{2x}}{5^x} = 5^{2x-x} = 5^x$ . Remember: multiplication = add, division = subtract!

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