Compare Algebraic Fractions: Evaluating (ab/xy)¹⁰ vs Complex Fraction Expression

Q: Why are these two expressions equal when they look so different?

They look different but use the same exponent rules! The left side has exponents already distributed, while the right side has the exponent applied to the entire fraction. Both equal $ \frac{a^{10}b^{10}}{x^{10}y^{10}} $.

Q: How do I know when to distribute exponents?

Use the power rule: $ (xy)^n = x^n \cdot y^n $ and $ (\frac{x}{y})^n = \frac{x^n}{y^n} $. When you see an exponent outside parentheses, distribute it to each factor inside.

Q: What if the variables have different signs?

The equality still holds regardless of whether the variables are positive or negative! Exponent rules work the same way with all real numbers (except division by zero).

Q: Can I simplify this further?

Both expressions are already in their simplest form. You could factor out common terms if specific values were given, but with variables, this is as simplified as it gets.

Q: How do I verify my comparison is correct?

Simplify both sides separately using exponent rules, then compare the results. If they're identical expressions, use the equals sign (=). If not, determine which is larger.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Select the appropriate sign

00:04 According to the laws of exponents, a fraction raised to the power (N)

00:07 equals a fraction where both the numerator and denominator are raised to the power (N)

00:10 We will apply this formula to our exercise

00:14 Raise both the numerator and denominator to the appropriate power, maintaining the parentheses

00:19 According to the laws of exponents, a product raised to the power (N)

00:23 is equal to the product broken down into factors where each factor is raised to the power (N)

00:27 We will apply this formula to our exercise, in the opposite direction

00:30 We'll combine the factors into one product inside of parentheses and raise them to the power

00:37 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the compatible sign:

$>,<,=$

$\frac{a^{10}\times b^{10}}{x^{10}\times y^{10}}\Box\left(\frac{a\times b}{x\times y}\right)^{10}$

2

Step-by-step solution

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

Step 1: Simplify both expressions using the properties of exponents.
Step 2: Compare the simplified results to determine the correct sign.

Now, let us simplify each side:

Step 1: Simplifying the left-hand side:
$\frac{a^{10} \times b^{10}}{x^{10} \times y^{10}}$

This expression can be expanded as:
$\frac{a^{10} b^{10}}{x^{10} y^{10}} = \left(\frac{a^{10}}{x^{10}}\right) \times \left(\frac{b^{10}}{y^{10}}\right)$
$= \left(\frac{a}{x}\right)^{10} \times \left(\frac{b}{y}\right)^{10}$

Step 2: Simplifying the right-hand side:
$\left(\frac{a \times b}{x \times y}\right)^{10}$

This can be expressed as:
$\left( \frac{a \times b}{x \times y} \right)^{10} = \left( \frac{a}{x} \times \frac{b}{y} \right)^{10}$
With the rule of exponentiating a product, this becomes:
$= \left(\frac{a}{x}\right)^{10} \times \left(\frac{b}{y}\right)^{10}$

After simplifying both expressions, we can see:
Both the left-hand side and the right-hand side are identical:

\left(\frac{a}{x}\right)^{10} \times \left(\frac{b}{y}\right)^{10} = \left(\frac{a}{x}\right)^{10} \times \left(\frac{b}{y}\right)^{10}

Therefore, the correct sign to use is $=$ .

The comparative relationship between the expressions is equal.

Thus, the solution to the problem is $=$ .

3

Final Answer

=

Key Points to Remember

Essential concepts to master this topic

Rule: Power of a product equals product of powers
Technique: $(\frac{ab}{xy})^{10} = \frac{a^{10}b^{10}}{x^{10}y^{10}}$ using exponent distribution
Check: Both sides simplify to identical expressions so they're equal ✓

Common Mistakes

Avoid these frequent errors

Applying exponents incorrectly to fraction components
Don't apply the exponent to only the numerator or denominator = unequal expressions! This ignores the power rule for fractions and gives wrong comparisons. Always apply the exponent to both numerator and denominator when dealing with fraction powers.

Practice Quiz

Test your knowledge with interactive questions

$$ Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

FAQ

Everything you need to know about this question

Why are these two expressions equal when they look so different?

+

They look different but use the same exponent rules! The left side has exponents already distributed, while the right side has the exponent applied to the entire fraction. Both equal $\frac{a^{10}b^{10}}{x^{10}y^{10}}$ .

How do I know when to distribute exponents?

+

Use the power rule: $(xy)^n = x^n \cdot y^n$ and $(\frac{x}{y})^n = \frac{x^n}{y^n}$ . When you see an exponent outside parentheses, distribute it to each factor inside.

What if the variables have different signs?

+

The equality still holds regardless of whether the variables are positive or negative! Exponent rules work the same way with all real numbers (except division by zero).

Can I simplify this further?

+

Both expressions are already in their simplest form. You could factor out common terms if specific values were given, but with variables, this is as simplified as it gets.

How do I verify my comparison is correct?

+

Simplify both sides separately using exponent rules, then compare the results. If they're identical expressions, use the equals sign (=). If not, determine which is larger.

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Compare Algebraic Fractions: Evaluating (ab/xy)¹⁰ vs Complex Fraction Expression

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