Complete the Expression: (5/7)^ax Power Formula

Q: Why can't I just multiply 5 and 7 by ax?

Because exponentiation and multiplication are different operations! When you see $ \left(\frac{5}{7}\right)^{ax} $, the exponent ax applies to the entire fraction, not as a multiplier.

Q: What's the difference between the correct and incorrect answers?

The correct answer $ \frac{5^{ax}}{7^{ax}} $ shows both parts raised to the power. Wrong answers like $ \frac{5^{ax}}{7} $ only raise one part, which completely changes the mathematical meaning.

Q: How do I remember this exponent rule?

Think of it as: "When a fraction gets an exponent, both the top AND bottom get that same exponent." The rule $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $ applies the power to each part separately.

Q: Does this work with any exponent?

Yes! Whether the exponent is a number, variable, or expression like ax, the rule stays the same. The exponent always applies to both numerator and denominator.

Q: What if the exponent is negative?

The same rule applies! $ \left(\frac{5}{7}\right)^{-ax} = \frac{5^{-ax}}{7^{-ax}} $. Remember that negative exponents create reciprocals, but the fundamental rule doesn't change.

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\left(\frac{5}{7}\right)^{ax}=$

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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, a fraction raised to the power (N)

00:06 equals the numerator and denominator raised to the same power (N)

00:11 We will apply this formula to our exercise

00:16 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:

$\left(\frac{5}{7}\right)^{ax}=$

Step-by-step solution

To solve the problem, follow these steps:

Identify the given expression: $\left(\frac{5}{7}\right)^{ax}$ .
Apply the rule for exponentiation of a fraction: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .
Exponentiate both the numerator and the denominator separately by $ax$ :
Calculate $\frac{5^{ax}}{7^{ax}}$ which follows directly from the application of the property.

Therefore, the rewritten expression is $\frac{5^{ax}}{7^{ax}}$ .

Among the given choices, the correct one is:

Choice 4: $\frac{5^{ax}}{7^{ax}}$

Final Answer

$\frac{5^{ax}}{7^{ax}}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising fractions to powers, exponentiate numerator and denominator separately
Technique: Apply $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to get $\frac{5^{ax}}{7^{ax}}$
Check: Verify the exponent applies to both top and bottom parts equally ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to the numerator
Don't write $\frac{5^{ax}}{7}$ when you have $\left(\frac{5}{7}\right)^{ax}$ = wrong mathematical operation! This ignores that the entire fraction is being raised to the power. Always apply the exponent to both the numerator AND denominator separately.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just multiply 5 and 7 by ax?

Because exponentiation and multiplication are different operations! When you see $\left(\frac{5}{7}\right)^{ax}$ , the exponent ax applies to the entire fraction, not as a multiplier.

What's the difference between the correct and incorrect answers?

The correct answer $\frac{5^{ax}}{7^{ax}}$ shows both parts raised to the power. Wrong answers like $\frac{5^{ax}}{7}$ only raise one part, which completely changes the mathematical meaning.

How do I remember this exponent rule?

Think of it as: "When a fraction gets an exponent, both the top AND bottom get that same exponent." The rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ applies the power to each part separately.

Does this work with any exponent?

Yes! Whether the exponent is a number, variable, or expression like ax, the rule stays the same. The exponent always applies to both numerator and denominator.

What if the exponent is negative?

The same rule applies! $\left(\frac{5}{7}\right)^{-ax} = \frac{5^{-ax}}{7^{-ax}}$ . Remember that negative exponents create reciprocals, but the fundamental rule doesn't change.

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