Simplify the Expression: 7^(x+1) × 7^x Using Exponent Rules

Q: Why do I add the exponents instead of multiplying them?

The multiplication rule for exponents states: $ a^m \times a^n = a^{m+n} $. Think of it as $ 7^{x+1} $ means 7 multiplied by itself (x+1) times, and $ 7^x $ means 7 multiplied by itself x more times!

Q: What's the difference between this and the power rule?

The power rule $ (a^m)^n = a^{mn} $ applies when you have parentheses, like $ (7^x)^{x+1} $. Here we're multiplying two separate exponential expressions, so we use the multiplication rule instead.

Q: How do I simplify (x+1) + x?

Just combine like terms! $ (x+1) + x = x + 1 + x = 2x + 1 $. Remember to distribute any negative signs and collect all the x terms together.

Q: Can I check my answer with specific numbers?

Absolutely! Try x = 1: $ 7^{1+1} \times 7^1 = 7^2 \times 7^1 = 49 \times 7 = 343 $. Using our answer: $ 7^{2(1)+1} = 7^3 = 343 $ ✓

Q: What if the bases were different numbers?

If the bases are different (like $ 7^x \times 3^x $), you cannot use this rule! The multiplication rule for exponents only works when the bases are identical.

Exponent Rules with Same Base Multiplication

$7^{x+1}\times7^x=$

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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)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:11 We will apply this formula to our exercise

00:15 We'll maintain the base and add the exponents together

00:23 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$7^{x+1}\times7^x=$

Step-by-step solution

To solve the problem $7^{x+1}\times7^x$ , follow these steps:

Step 1: Use the rule for multiplying powers with the same base:

$a^m \cdot a^n = a^{m+n}$

Here, the base $a$ is $7$ , and the exponents are $x+1$ and $x$ .

Step 2: Add the exponents:

The expression becomes $7^{(x+1) + x}$ .

Step 3: Simplify the exponents:

$(x+1) + x = 2x + 1$

Step 4: Write the final expression:

The simplified expression is $7^{2x+1}$ .

Therefore, our solution matches choice 3.

The solution to the problem is $7^{2x+1}$ .

Final Answer

$7^{2x+1}$

Key Points to Remember

Essential concepts to master this topic

Base Rule: When multiplying powers with same base, add exponents
Technique: $7^{x+1} \times 7^x = 7^{(x+1)+x} = 7^{2x+1}$
Check: Verify by substituting a test value like x=2: $7^3 \times 7^2 = 7^5$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply (x+1) × x = x(x+1) to get $7^{x(x+1)}$ ! This confuses the multiplication rule with the power rule and gives a completely wrong answer. Always add exponents when multiplying powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

The multiplication rule for exponents states: $a^m \times a^n = a^{m+n}$ . Think of it as $7^{x+1}$ means 7 multiplied by itself (x+1) times, and $7^x$ means 7 multiplied by itself x more times!

What's the difference between this and the power rule?

The power rule $(a^m)^n = a^{mn}$ applies when you have parentheses, like $(7^x)^{x+1}$ . Here we're multiplying two separate exponential expressions, so we use the multiplication rule instead.

How do I simplify (x+1) + x?

Just combine like terms! $(x+1) + x = x + 1 + x = 2x + 1$ . Remember to distribute any negative signs and collect all the x terms together.

Can I check my answer with specific numbers?

Absolutely! Try x = 1: $7^{1+1} \times 7^1 = 7^2 \times 7^1 = 49 \times 7 = 343$ . Using our answer: $7^{2(1)+1} = 7^3 = 343$ ✓

What if the bases were different numbers?

If the bases are different (like $7^x \times 3^x$ ), you cannot use this rule! The multiplication rule for exponents only works when the bases are identical.

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