Calculate (5/8)^9: Evaluating a Fraction Raised to the Ninth Power

Q: Do I need to calculate the actual values of 5⁹ and 8⁹?

Not necessarily! The question asks for the expression, not the numerical value. $ \frac{5^9}{8^9} $ is the correct mathematical form, even without calculating that 5⁹ = 1,953,125.

Q: What if I see a negative exponent like (5/8)⁻³?

For negative exponents, flip the fraction first! $ \left(\frac{5}{8}\right)^{-3} = \left(\frac{8}{5}\right)^3 = \frac{8^3}{5^3} $. The negative sign means "take the reciprocal."

Q: Can I simplify 5⁹/8⁹ further?

Only if 5 and 8 share common factors, but since 5 and 8 are relatively prime (no common factors except 1), $ \frac{5^9}{8^9} $ is already in simplest form!

Q: Is there a shortcut for large exponents?

The rule $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $ works for any exponent, no matter how big! Just apply it directly - no shortcuts needed, just the power distribution rule.

Fraction Exponents with Power Distribution Rule

Insert the corresponding expression:

$\left(\frac{5}{8}\right)^9=$

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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, a fraction raised to a power (N)

00:07 equals the numerator and denominator raised to the same power (N)

00:10 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:

$\left(\frac{5}{8}\right)^9=$

Step-by-step solution

To solve this problem, we'll apply the rule for raising a fraction to a power:

Using the formula $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ , we can express $\left(\frac{5}{8}\right)^9$ as follows:

Step 1: Identify the base and exponent in $\left(\frac{5}{8}\right)^9$ . Here, $a = 5$ , $b = 8$ , and $n = 9$ .

Step 2: Apply the exponentiation rule:
$\left(\frac{5}{8}\right)^9 = \frac{5^9}{8^9}$ .

Therefore, the original expression simplifies to $\frac{5^9}{8^9}$ .

As a result, the correct rewritten form of $\left(\frac{5}{8}\right)^9$ is $\frac{5^9}{8^9}$ .

Final Answer

$\frac{5^9}{8^9}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{5}{8}\right)^9 = \frac{5^9}{8^9}$ distributes the 9
Check: Verify that both 5 and 8 have the same exponent ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to one part of the fraction
Don't write $\left(\frac{5}{8}\right)^9 = \frac{5^9}{8}$ = wrong expression! The exponent must apply to the entire fraction, affecting both numerator and denominator. Always distribute the exponent: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why does the exponent apply to both the top and bottom?

When you raise a fraction to a power, you're multiplying the entire fraction by itself that many times. Since $\frac{5}{8} \times \frac{5}{8} = \frac{5 \times 5}{8 \times 8}$ , doing this 9 times gives $\frac{5^9}{8^9}$ !

Do I need to calculate the actual values of 5⁹ and 8⁹?

Not necessarily! The question asks for the expression, not the numerical value. $\frac{5^9}{8^9}$ is the correct mathematical form, even without calculating that 5⁹ = 1,953,125.

What if I see a negative exponent like (5/8)⁻³?

For negative exponents, flip the fraction first! $\left(\frac{5}{8}\right)^{-3} = \left(\frac{8}{5}\right)^3 = \frac{8^3}{5^3}$ . The negative sign means "take the reciprocal."

Can I simplify 5⁹/8⁹ further?

Only if 5 and 8 share common factors, but since 5 and 8 are relatively prime (no common factors except 1), $\frac{5^9}{8^9}$ is already in simplest form!

Is there a shortcut for large exponents?

The rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ works for any exponent, no matter how big! Just apply it directly - no shortcuts needed, just the power distribution rule.

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