Simplify the Expression: 7^(x+a) × 7^a × 7^x

Q: Why do we add exponents instead of multiplying them?

The product rule for exponents states that $ a^m \times a^n = a^{m+n} $. We add because multiplication of powers means repeated multiplication of the base.

Q: What if the base numbers were different?

If bases are different (like $ 7^x \times 5^x $), you cannot combine them using exponent rules. The bases must be identical to use the product rule.

Q: How do I handle parentheses in exponents like (x+a)?

Treat the entire expression in parentheses as one unit. So $ 7^{x+a} $ has exponent (x+a), and you add this whole thing to other exponents.

Q: Can I factor out common terms from the final answer?

Yes! $ 7^{2x+2a} $ can also be written as $ 7^{2(x+a)} $ by factoring out 2 from the exponent, but both forms are correct.

Exponent Rules with Multiple Terms

Reduce the following equation:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, the multiplication of exponents with equal bases (A)

00:08 equals the same base raised to the sum of the exponents (N+M)

00:13 We will apply this formula to our exercise

00:17 We'll maintain the base and add the exponents together

00:44 Let's group the factors together

00:47 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:

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

Step-by-step solution

To solve this problem, we'll start by applying the rules for multiplying powers with the same base.

Step 1: Identify the expression given as $7^{x+a} \times 7^a \times 7^x$ .
Step 2: Apply the rule $a^m \times a^n = a^{m+n}$ to combine the exponents since all terms have the same base, 7.

Let's proceed to simplify:

Combine the exponents:

$7^{x+a} \times 7^a \times 7^x = 7^{(x+a) + a + x}$ .

Now, simplify the addition of the exponents:

$7^{(x+a) + a + x} = 7^{x + a + a + x}$ .

Combine like terms in the exponent:

$x + a + a + x = 2x + 2a$ .

Thus, the expression simplifies to:

$7^{2x+2a}$ .

Therefore, the simplification of the given expression is $7^{2x+2a}$ .

Final Answer

$7^{2x+2a}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add exponents
Technique: $7^{x+a} \times 7^a \times 7^x = 7^{(x+a)+a+x}$
Check: Combine like terms in exponent: x+a+a+x = 2x+2a ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply exponents like (x+a)×a×x = wrong answer! This confuses the power rule with the product rule. Always add exponents when multiplying powers with the same base.

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 instead of multiplying them?

The product rule for exponents states that $a^m \times a^n = a^{m+n}$ . We add because multiplication of powers means repeated multiplication of the base.

What if the base numbers were different?

If bases are different (like $7^x \times 5^x$ ), you cannot combine them using exponent rules. The bases must be identical to use the product rule.

How do I handle parentheses in exponents like (x+a)?

Treat the entire expression in parentheses as one unit. So $7^{x+a}$ has exponent (x+a), and you add this whole thing to other exponents.

Can I factor out common terms from the final answer?

Yes! $7^{2x+2a}$ can also be written as $7^{2(x+a)}$ by factoring out 2 from the exponent, but both forms are correct.

What if there were four or more terms to multiply?

The same rule applies! Keep adding all the exponents together. For example: $7^a \times 7^b \times 7^c \times 7^d = 7^{a+b+c+d}$

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