Simplify the Expression: 5²ᵃ × 8²ᵃ × 7²ᵃ Using Laws of Exponents

Q: Why can I combine the bases when the exponents are the same?

This works because of the power of a product rule! When you have $ (abc)^n $, it equals $ a^n \cdot b^n \cdot c^n $. We're just using this rule in reverse.

Q: What if the exponents were different, like $ 5^{2a} \times 8^{3a} $ ?

You cannot combine bases when exponents are different! The expression would stay as $ 5^{2a} \times 8^{3a} $ because there's no exponent rule that applies.

Q: Do I need to calculate $ 5 \times 8 \times 7 $ ?

Usually not! The answer $ (5 \times 8 \times 7)^{2a} $ is perfectly acceptable. Only calculate $ 5 \times 8 \times 7 = 280 $ if specifically asked to simplify further.

Q: Can this method work with more than three terms?

Absolutely! You can combine any number of terms with the same exponent: $ a^n \cdot b^n \cdot c^n \cdot d^n = (abcd)^n $. The rule works for any amount of bases.

Q: What about negative exponents like $ 5^{-2a} \times 8^{-2a} $ ?

The same rule applies! $ 5^{-2a} \times 8^{-2a} = (5 \times 8)^{-2a} $. The sign of the exponent doesn't matter - only that they're identical.

Exponent Laws with Common Powers

Insert the corresponding expression:

$5^{2a}\times8^{2a}\times7^{2a}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's simplify this problem together.

00:15 Remember, by the laws of exponents, when you raise a product to a power, like N,

00:21 each factor in the product is raised to that same power, N.

00:27 We'll use this rule for our exercise.

00:30 Notice, every factor is raised to power N here.

00:42 And there you have it, the problem is solved!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$5^{2a}\times8^{2a}\times7^{2a}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the common exponent in the expression.
Step 2: Use the power of a product rule to rewrite the expression with a single exponent.

Now, let's work through each step:

Step 1: Recognize that each term in the expression $5^{2a} \times 8^{2a} \times 7^{2a}$ has the same exponent, which is $2a$ .

Step 2: Apply the power of a product rule. This rule states that $(x \times y \times z)^n = x^n \cdot y^n \cdot z^n$ , which can be reversed to combine the terms.

With this understanding, the expression $5^{2a} \times 8^{2a} \times 7^{2a}$ can be rewritten as a single exponent expression by combining the bases:

$(5 \cdot 8 \cdot 7)^{2a}$ .

Therefore, the solution to this problem is $\left(5 \times 8 \times 7\right)^{2a}$ .

Final Answer

$\left(5\times8\times7\right)^{2a}$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases have same exponent, combine using $(abc)^n = a^n \cdot b^n \cdot c^n$
Technique: Reverse the rule: $5^{2a} \times 8^{2a} \times 7^{2a} = (5 \times 8 \times 7)^{2a}$
Check: Verify by expanding back: $(280)^{2a} = 280^{2a}$ matches original form ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of combining bases
Don't add the exponents to get $5^{6a} \times 8^{6a} \times 7^{6a}$ = wrong answer! This changes the problem completely and gives massive incorrect values. Always keep the exponent the same and combine only the bases when exponents are identical.

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why can I combine the bases when the exponents are the same?

This works because of the power of a product rule! When you have $(abc)^n$ , it equals $a^n \cdot b^n \cdot c^n$ . We're just using this rule in reverse.

What if the exponents were different, like $5^{2a} \times 8^{3a}$ ?

You cannot combine bases when exponents are different! The expression would stay as $5^{2a} \times 8^{3a}$ because there's no exponent rule that applies.

Do I need to calculate $5 \times 8 \times 7$ ?

Usually not! The answer $(5 \times 8 \times 7)^{2a}$ is perfectly acceptable. Only calculate $5 \times 8 \times 7 = 280$ if specifically asked to simplify further.

Can this method work with more than three terms?

Absolutely! You can combine any number of terms with the same exponent: $a^n \cdot b^n \cdot c^n \cdot d^n = (abcd)^n$ . The rule works for any amount of bases.

What about negative exponents like $5^{-2a} \times 8^{-2a}$ ?

The same rule applies! $5^{-2a} \times 8^{-2a} = (5 \times 8)^{-2a}$ . The sign of the exponent doesn't matter - only that they're identical.

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Simplify the Expression: 5²ᵃ × 8²ᵃ × 7²ᵃ Using Laws of Exponents

Exponent Laws with Common Powers

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can I combine the bases when the exponents are the same?

What if the exponents were different, like $5^{2a} \times 8^{3a}$ ?

Do I need to calculate $5 \times 8 \times 7$ ?

Can this method work with more than three terms?

What about negative exponents like $5^{-2a} \times 8^{-2a}$ ?

🌟 Unlock Your Math Potential

More Questions

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Simplify the Expression: 5^(y+1) × 3^(y+1) Product Rule Challenge

Calculate (9×7×6×2)^6: Complex Product Exponentiation

Simplify the Product: 3^x × 7^x × 5^x Using Exponent Rules

Solve: 4^a × 8^a × 2^a - Multiplying Powers with Same Exponent

Simplify the Expression: 5²ᵃ × 8²ᵃ × 7²ᵃ Using Laws of Exponents

Exponent Laws with Common Powers

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can I combine the bases when the exponents are the same?

What if the exponents were different, like 52a×83a 5^{2a} \times 8^{3a} 52a×83a?

Do I need to calculate 5×8×7 5 \times 8 \times 7 5×8×7?

Can this method work with more than three terms?

What about negative exponents like 5−2a×8−2a 5^{-2a} \times 8^{-2a} 5−2a×8−2a?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Simplify the Expression: 5^(y+1) × 3^(y+1) Product Rule Challenge

Calculate (9×7×6×2)^6: Complex Product Exponentiation

Simplify the Product: 3^x × 7^x × 5^x Using Exponent Rules

Solve: 4^a × 8^a × 2^a - Multiplying Powers with Same Exponent

What if the exponents were different, like $5^{2a} \times 8^{3a}$ ?

Do I need to calculate $5 \times 8 \times 7$ ?

What about negative exponents like $5^{-2a} \times 8^{-2a}$ ?