Answer :
To determine which expressions are equivalent to the given expression [tex]\( y^{-8} y^3 x^0 x^{-2} \)[/tex], we need to simplify it using the properties of exponents. Here is the step-by-step process:
1. Combine the terms with the same base:
- For the [tex]\(y\)[/tex] terms: [tex]\( y^{-8} \cdot y^3 \)[/tex]
Using the property of exponents [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]:
[tex]\[ y^{-8} \cdot y^3 = y^{-8 + 3} = y^{-5} \][/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\( x^0 \cdot x^{-2} \)[/tex]
Using the property of exponents [tex]\( a^0 = 1 \)[/tex] for any non-zero [tex]\(a\)[/tex]:
[tex]\[ x^0 = 1 \][/tex]
Therefore,
[tex]\[ x^0 \cdot x^{-2} = 1 \cdot x^{-2} = x^{-2} \][/tex]
2. Combine the simplified terms:
[tex]\[ y^{-5} \cdot x^{-2} = x^{-2} \cdot y^{-5} \][/tex]
The simplified expression is [tex]\( x^{-2} \cdot y^{-5} \)[/tex].
Now, let's check if any of the given expressions are equivalent to [tex]\( x^{-2} \cdot y^{-5} \)[/tex]:
1. [tex]\( \frac{1}{y^{2 x}} \)[/tex]
- This is not equivalent because it has a completely different form and involves different exponents.
2. [tex]\( \frac{g^2}{y^{11}} \)[/tex]
- This is not equivalent because it involves an entirely different base [tex]\(g\)[/tex] and different exponents.
3. [tex]\( y^{-24} \)[/tex]
- This is not equivalent because it only involves [tex]\( y \)[/tex] and the exponent does not match [tex]\( -5 \)[/tex].
4. [tex]\( x^2 y^{-11} \)[/tex]
- This is not equivalent because the exponents of both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] do not match [tex]\( -2 \)[/tex] and [tex]\( -5 \)[/tex], respectively.
5. [tex]\( x^{-2} y^{-5} \)[/tex]
- This matches exactly with our simplified expression and is, therefore, equivalent.
6. [tex]\( \frac{1}{x^2 y^5} \)[/tex]
- This is not equivalent because the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are not negative.
The correct equivalent expression is:
[tex]\[ x^{-2} y^{-5} \][/tex]
1. Combine the terms with the same base:
- For the [tex]\(y\)[/tex] terms: [tex]\( y^{-8} \cdot y^3 \)[/tex]
Using the property of exponents [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]:
[tex]\[ y^{-8} \cdot y^3 = y^{-8 + 3} = y^{-5} \][/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\( x^0 \cdot x^{-2} \)[/tex]
Using the property of exponents [tex]\( a^0 = 1 \)[/tex] for any non-zero [tex]\(a\)[/tex]:
[tex]\[ x^0 = 1 \][/tex]
Therefore,
[tex]\[ x^0 \cdot x^{-2} = 1 \cdot x^{-2} = x^{-2} \][/tex]
2. Combine the simplified terms:
[tex]\[ y^{-5} \cdot x^{-2} = x^{-2} \cdot y^{-5} \][/tex]
The simplified expression is [tex]\( x^{-2} \cdot y^{-5} \)[/tex].
Now, let's check if any of the given expressions are equivalent to [tex]\( x^{-2} \cdot y^{-5} \)[/tex]:
1. [tex]\( \frac{1}{y^{2 x}} \)[/tex]
- This is not equivalent because it has a completely different form and involves different exponents.
2. [tex]\( \frac{g^2}{y^{11}} \)[/tex]
- This is not equivalent because it involves an entirely different base [tex]\(g\)[/tex] and different exponents.
3. [tex]\( y^{-24} \)[/tex]
- This is not equivalent because it only involves [tex]\( y \)[/tex] and the exponent does not match [tex]\( -5 \)[/tex].
4. [tex]\( x^2 y^{-11} \)[/tex]
- This is not equivalent because the exponents of both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] do not match [tex]\( -2 \)[/tex] and [tex]\( -5 \)[/tex], respectively.
5. [tex]\( x^{-2} y^{-5} \)[/tex]
- This matches exactly with our simplified expression and is, therefore, equivalent.
6. [tex]\( \frac{1}{x^2 y^5} \)[/tex]
- This is not equivalent because the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are not negative.
The correct equivalent expression is:
[tex]\[ x^{-2} y^{-5} \][/tex]
Answer:y
−8
⋅y
3
⋅x
0
⋅x
−2
Step-by-step explanation:
combine the powers of
y:
−
8
⋅
3
=
−
8
+
3
=
−
5
y
−8
⋅y
3
=y
−8+3
=y
−5
Simplify
0
x
0
:
Any number raised to the power of 0 is 1, so
0
=
1
x
0
=1.
Combine with
−
2
x
−2
:
1
⋅
−
2
=
−
2
1⋅x
−2
=x
−2
So, the expression simplifies to:
−
5
⋅
−
2
=
1
5
⋅
1
2
=
1
5
2
y
−5
⋅x
−2
=
y
5
1
⋅
x
2
1
=
y
5
x
2
1
Equivalent expressions:
1
5
2
y
5
x
2
1
−
2
−
5
x
−2
y
−5
Non-equivalent expressions:
−
11
−
2
y
−11
x
−2
(incorrect, since powers of
y are not added correctly)
−
11
2
x
2
y
−11
(incorrect, similar issue with powers)
−
5
2
y
−5
x
2
(incorrect, as the power of
x should be negative)
−
2
−
5
y
−5
x
−2
(incorrect due to incorrect placement of powers in the denominator)
So, the correct answers are:
−
5
−
2
y
−5
x
−2
1
5
2
y
5
x
2
1
