Simplify the Expression: 8^a × 8^2 × 8^x Using Laws of Exponents

Q: Why do we add exponents when multiplying?

Think of it this way: $ 8^2 \times 8^3 $ means (8×8) × (8×8×8) = 8×8×8×8×8 = $ 8^5 $. We're counting total factors: 2 + 3 = 5!

Q: What if the bases are different?

The rule only works with identical bases. For $ 2^3 \times 8^2 $, you can't combine directly. You'd need to rewrite with the same base first (like $ 2^3 \times (2^3)^2 $).

Q: Can I use this rule with variables in the exponents?

Absolutely! Variables like a and x in exponents work the same way. Just add them algebraically: a + 2 + x becomes your new exponent.

Q: What about when I'm dividing powers?

For division, you subtract exponents instead! $ 8^5 ÷ 8^2 = 8^{5-2} = 8^3 $. Remember: multiply = add, divide = subtract.

Exponent Laws with Multiple Terms

Reduce the following equation:

$8^a\times8^2\times8^x=$

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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, multiplying exponents with the same base (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

00:24 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:

$8^a\times8^2\times8^x=$

Step-by-step solution

To solve this problem, we'll use the property of exponents for multiplying powers with the same base:

Step 1: Identify that all terms have the same base, which is $8$ . The equation is given as $8^a \times 8^2 \times 8^x$ .
Step 2: Apply the multiplication property of exponents: $b^m \times b^n = b^{m+n}$ .
Step 3: Add the exponents: $(a) + (2) + (x)$ to get the new exponent for the single base.

By applying these steps, we obtain:

$8^{a+2+x}$

This result matches choice 1, confirming that this is the correct simplified expression.

Final Answer

$8^{a+2+x}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: For $8^a \times 8^2 \times 8^x$ , combine as $8^{a+2+x}$
Check: Verify by substituting values: if a=1, x=3, then $8^1 \times 8^2 \times 8^3 = 8^6$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply exponents like $8^{a \times 2 \times x} = 8^{2ax}$ ! This gives a completely different (and wrong) answer. The multiplication rule only applies to the bases, not exponents. Always add the 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 we add exponents when multiplying?

Think of it this way: $8^2 \times 8^3$ means (8×8) × (8×8×8) = 8×8×8×8×8 = $8^5$ . We're counting total factors: 2 + 3 = 5!

What if the bases are different?

The rule only works with identical bases. For $2^3 \times 8^2$ , you can't combine directly. You'd need to rewrite with the same base first (like $2^3 \times (2^3)^2$ ).

Can I use this rule with variables in the exponents?

Absolutely! Variables like a and x in exponents work the same way. Just add them algebraically: a + 2 + x becomes your new exponent.

What about when I'm dividing powers?

For division, you subtract exponents instead! $8^5 ÷ 8^2 = 8^{5-2} = 8^3$ . Remember: multiply = add, divide = subtract.

How can I double-check my answer?

Try substituting simple numbers for the variables! If a=1 and x=3, then $8^1 \times 8^2 \times 8^3 = 8 \times 64 \times 512$ should equal $8^{1+2+3} = 8^6$ .

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