Select all the correct answers.
Which expressions are equivalent to the given expression?
[tex]$y^{-8} y^3 x^0 x^{-2}$[/tex]
[tex]$x^2 y^{-11}$[/tex]
[tex]$\frac{1}{x^2 y^5}$[/tex]
[tex]$\frac{x^2}{y^{11}}$[/tex]
[tex]$y^{-24}$[/tex]
[tex]$\frac{1}{y^{24}}$[/tex]
[tex]$x^{-2} y^{-5}$[/tex]
Answer (1)
The given expression y − 8 y 3 x 0 x − 2 simplifies to x − 2 y − 5 . This is equivalent to x 2 y 5 1 . Therefore, the equivalent expressions from the list are:
x 2 y 5 1
x − 2 y − 5
The final answer is x 2 y 5 1 , x − 2 y − 5 .
Explanation
Understanding the Problem We are given the expression y − 8 y 3 x 0 x − 2 and we want to find equivalent expressions from the list.
Simplifying the Expression First, let's simplify the given expression using the properties of exponents. Recall that x 0 = 1 for any non-zero x , and x a x b = x a + b .
Combining the Terms We can simplify the y terms as follows: y − 8 y 3 = y − 8 + 3 = y − 5 And the x terms as follows: x 0 x − 2 = 1 ⋅ x − 2 = x − 2 So the simplified expression is x − 2 y − 5 .
Rewriting with Positive Exponents Now, let's rewrite the simplified expression using positive exponents. Recall that x − a = x a 1 .
x − 2 y − 5 = x 2 1 ⋅ y 5 1 = x 2 y 5 1 So, x − 2 y − 5 is equivalent to x 2 y 5 1 .
Identifying Equivalent Expressions Now, let's compare our simplified expression x − 2 y − 5 or x 2 y 5 1 with the list of expressions given:
x 2 y − 11 - Not equivalent
x 2 y 5 1 - Equivalent
y 11 x 2 - Not equivalent
y − 24 - Not equivalent
y 24 1 - Not equivalent
x − 2 y − 5 - Equivalent
Final Answer Therefore, the expressions equivalent to y − 8 y 3 x 0 x − 2 are x 2 y 5 1 and x − 2 y − 5 .
Examples
Understanding how to simplify expressions with exponents is crucial in many areas, such as calculating the area and volume of geometric shapes. For example, if you have a rectangle with sides of length x − 2 y and x y − 3 , the area would be ( x − 2 y ) ( x y − 3 ) = x − 1 y − 2 = x y 2 1 . This skill is also essential in physics when dealing with units and scientific notation.
