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

Insert the compatible sign:

>,<,= >,<,=

a10×b10x10×y10(a×bx×y)10 \frac{a^{10}\times b^{10}}{x^{10}\times y^{10}}\Box\left(\frac{a\times b}{x\times y}\right)^{10}

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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:

>,<,= >,<,=

a10×b10x10×y10(a×bx×y)10 \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:
a10×b10x10×y10\frac{a^{10} \times b^{10}}{x^{10} \times y^{10}}

This expression can be expanded as:
a10b10x10y10=(a10x10)×(b10y10) \frac{a^{10} b^{10}}{x^{10} y^{10}} = \left(\frac{a^{10}}{x^{10}}\right) \times \left(\frac{b^{10}}{y^{10}}\right)
=(ax)10×(by)10= \left(\frac{a}{x}\right)^{10} \times \left(\frac{b}{y}\right)^{10}

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

This can be expressed as:
(a×bx×y)10=(ax×by)10 \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:
=(ax)10×(by)10 = \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:

(ax)10×(by)10=(ax)10×(by)10 \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

=

Practice Quiz

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\( 112^0=\text{?} \)

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