Simplify the Exponential Fraction: (5¹⁰ × 8¹⁰)/(4¹⁰ × 7¹⁰)

Q: Why can I combine numbers with the same exponent?

The exponent property $ a^n \times b^n = (a \times b)^n $ works because you're multiplying the same number of factors. For example: $ 5^3 \times 8^3 = (5×5×5) \times (8×8×8) = (5×8)^3 $

Q: Do I have to simplify both parts at once?

Yes! For the cleanest result, apply the same exponent rule to both numerator and denominator. This gives you $ \frac{(5×8)^{10}}{(4×7)^{10}} $, which is the most simplified form.

Q: What if the exponents were different?

If exponents don't match, you cannot use this grouping rule! The property $ a^n \times b^n = (a×b)^n $ only works when both terms have the same exponent.

Q: Can I calculate the final numerical answer?

You could compute $ \frac{40^{10}}{28^{10}} $, but the algebraic form $ \frac{(5×8)^{10}}{(4×7)^{10}} $ is usually the preferred answer as it clearly shows the mathematical structure.

Q: Is there another way to write this?

Yes! You can also write it as $ \left(\frac{5×8}{4×7}\right)^{10} = \left(\frac{40}{28}\right)^{10} $ using the property $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $

Exponent Properties with Multiplication Grouping

Insert the corresponding expression:

$\frac{5^{10}\times8^{10}}{4^{10}\times7^{10}}=$

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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 exponent laws, a product raised to the power (N)

00:07 equals the product broken down into factors with each factor raised to the power (N)

00:11 We'll apply this formula to our exercise, converting to parentheses with an exponent

00:20 According to the exponent laws, a fraction raised to a power (N)

00:24 equals the numerator and denominator, each raised to the same power (N)

00:29 Now we'll apply the second formula and convert the expression to a fraction within parentheses with an exponent

00:34 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{5^{10}\times8^{10}}{4^{10}\times7^{10}}=$

Step-by-step solution

To simplify the expression $\frac{5^{10}\times8^{10}}{4^{10}\times7^{10}}$ , we start by applying the property of exponents: $(a \times b)^n = a^n \times b^n$ .

Step 1: Rewrite the expression. Notice that both the numerator and the denominator consist of two numbers, each raised to the power of 10.

$\frac{5^{10} \times 8^{10}}{4^{10} \times 7^{10}} = \frac{(5 \times 8)^{10}}{(4 \times 7)^{10}}$

Now, we notice we can apply the equality for exponential simplification:

$\left(\frac{5 \times 8}{4 \times 7}\right)^{10} = \frac{(5 \times 8)^{10}}{(4 \times 7)^{10}}$

Concluding, the simplified expression of the given problem is equivalent to option "1":

The expression simplifies to $\frac{\left(5\times8\right)^{10}}{\left(4\times7\right)^{10}}$ , which aligns perfectly with choice id="1".

Therefore, the final answer is:

$\frac{\left(5\times8\right)^{10}}{\left(4\times7\right)^{10}}$ .

Final Answer

$\frac{\left(5\times8\right)^{10}}{\left(4\times7\right)^{10}}$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases have same exponent, group before applying power
Technique: $a^n \times b^n = (a \times b)^n$ converts $5^{10} \times 8^{10} = (5 \times 8)^{10}$
Check: Both forms equal same value: $\frac{40^{10}}{28^{10}} = \frac{(5 \times 8)^{10}}{(4 \times 7)^{10}}$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent rule only to numerator or denominator
Don't combine just $5^{10} \times 8^{10} = (5 \times 8)^{10}$ while leaving $4^{10} \times 7^{10}$ separate = inconsistent simplification! This creates a mixed form that's harder to work with. Always apply the same exponent property to both numerator and denominator simultaneously.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine numbers with the same exponent?

The exponent property $a^n \times b^n = (a \times b)^n$ works because you're multiplying the same number of factors. For example: $5^3 \times 8^3 = (5×5×5) \times (8×8×8) = (5×8)^3$

Do I have to simplify both parts at once?

Yes! For the cleanest result, apply the same exponent rule to both numerator and denominator. This gives you $\frac{(5×8)^{10}}{(4×7)^{10}}$ , which is the most simplified form.

What if the exponents were different?

If exponents don't match, you cannot use this grouping rule! The property $a^n \times b^n = (a×b)^n$ only works when both terms have the same exponent.

Can I calculate the final numerical answer?

You could compute $\frac{40^{10}}{28^{10}}$ , but the algebraic form $\frac{(5×8)^{10}}{(4×7)^{10}}$ is usually the preferred answer as it clearly shows the mathematical structure.

Is there another way to write this?

Yes! You can also write it as $\left(\frac{5×8}{4×7}\right)^{10} = \left(\frac{40}{28}\right)^{10}$ using the property $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$

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