Answer :
- Apply the exponent rule $x^a
cdot x^b = x^{a+b}$ to simplify the $x$ terms: $3 x^4
cdot x^8 = 3x^{12}$.
- Rewrite the term with the negative exponent using the rule $y^{-n} =
\frac{1}{y^n}$: $y^{-2} =
\frac{1}{y^2}$.
- Combine the simplified terms: $3x^{12}
\cdot
\frac{1}{y^2} =
\frac{3x^{12}}{y^2}$.
- The final simplified expression is $\boxed{\frac{3 x^{12}}{y^2}}$.
### Explanation
1. Understanding the Expression
We are given the expression $3 x^4
cdot x^8 y^{-2}$. Our goal is to simplify this expression using the properties of exponents.
2. Simplifying the x terms
To simplify the expression, we'll use the rule that $x^a
cdot x^b = x^{a+b}$. Applying this rule to the $x$ terms, we have:
$$3 x^4
cdot x^8 y^{-2} = 3 x^{4+8} y^{-2} = 3 x^{12} y^{-2}$$
So, we add the exponents of the $x$ terms.
3. Simplifying the y term
Now, we need to deal with the $y^{-2}$ term. Recall that $y^{-n} =
\frac{1}{y^n}$. Therefore, we can rewrite $y^{-2}$ as $\frac{1}{y^2}$.
$$3 x^{12} y^{-2} = 3 x^{12}
\cdot
\frac{1}{y^2} =
\frac{3 x^{12}}{y^2}$$
This step involves rewriting the term with a negative exponent as a fraction.
4. Comparing with the options
Now we compare our simplified expression, $\frac{3 x^{12}}{y^2}$, with the given options:
A) $70 x^7$
B) $\frac{20 y^{20}}{x^2}$
C) $\frac{5 y^9}{x}$
D) $\frac{3 x^{12}}{y^2}$
We can see that option D matches our simplified expression.
5. Final Answer
Therefore, the correct answer is D) $\frac{3 x^{12}}{y^2}$.
### Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating the area or volume of geometric shapes, analyzing polynomial functions, and even in physics when dealing with scientific notation or units of measurement. For instance, if you are calculating the volume of a rectangular prism with sides that are expressed as variables with exponents, you would need to simplify the expression to find the volume. This skill is also crucial in computer science when analyzing algorithms and data structures.
cdot x^b = x^{a+b}$ to simplify the $x$ terms: $3 x^4
cdot x^8 = 3x^{12}$.
- Rewrite the term with the negative exponent using the rule $y^{-n} =
\frac{1}{y^n}$: $y^{-2} =
\frac{1}{y^2}$.
- Combine the simplified terms: $3x^{12}
\cdot
\frac{1}{y^2} =
\frac{3x^{12}}{y^2}$.
- The final simplified expression is $\boxed{\frac{3 x^{12}}{y^2}}$.
### Explanation
1. Understanding the Expression
We are given the expression $3 x^4
cdot x^8 y^{-2}$. Our goal is to simplify this expression using the properties of exponents.
2. Simplifying the x terms
To simplify the expression, we'll use the rule that $x^a
cdot x^b = x^{a+b}$. Applying this rule to the $x$ terms, we have:
$$3 x^4
cdot x^8 y^{-2} = 3 x^{4+8} y^{-2} = 3 x^{12} y^{-2}$$
So, we add the exponents of the $x$ terms.
3. Simplifying the y term
Now, we need to deal with the $y^{-2}$ term. Recall that $y^{-n} =
\frac{1}{y^n}$. Therefore, we can rewrite $y^{-2}$ as $\frac{1}{y^2}$.
$$3 x^{12} y^{-2} = 3 x^{12}
\cdot
\frac{1}{y^2} =
\frac{3 x^{12}}{y^2}$$
This step involves rewriting the term with a negative exponent as a fraction.
4. Comparing with the options
Now we compare our simplified expression, $\frac{3 x^{12}}{y^2}$, with the given options:
A) $70 x^7$
B) $\frac{20 y^{20}}{x^2}$
C) $\frac{5 y^9}{x}$
D) $\frac{3 x^{12}}{y^2}$
We can see that option D matches our simplified expression.
5. Final Answer
Therefore, the correct answer is D) $\frac{3 x^{12}}{y^2}$.
### Examples
Understanding how to simplify expressions with exponents is useful in many areas, such as calculating the area or volume of geometric shapes, analyzing polynomial functions, and even in physics when dealing with scientific notation or units of measurement. For instance, if you are calculating the volume of a rectangular prism with sides that are expressed as variables with exponents, you would need to simplify the expression to find the volume. This skill is also crucial in computer science when analyzing algorithms and data structures.
