Simplify The Expression:${ \left(3x^3 \left(y {-2}\right) 4\right) }$

Mar 11, 2025 by ADMIN 71 views

**Simplify the Expression: A Step-by-Step Guide** =====================================================

Understanding the Problem

The given expression is ${ \left(3x^3 \left(y^{-2}\right)4\right) }$. Our goal is to simplify this expression by applying the rules of exponents and algebra.

What are Exponents?

Exponents are a shorthand way of writing repeated multiplication. For example, $x^3$ means $x \times x \times x$ . When we have a negative exponent, it means we are taking the reciprocal of the base raised to the positive exponent. In this case, $y^{-2}$ means $\frac{1}{y^2}$ .

Simplifying the Expression

To simplify the expression, we need to apply the rules of exponents. The first step is to evaluate the exponent inside the parentheses.

Evaluating the Inner Exponent

The inner exponent is $y^{-2}$ . According to the rules of exponents, when we raise a power to a power, we multiply the exponents. Therefore, $y^{-2}$ can be rewritten as $\frac{1}{y^2}$ .

${ \left(3x^3 \left(y^{-2}\right)^4\right) = \left(3x^3 \left(\frac{1}{y^2}\right)^4\right) }$

Simplifying the Fraction

Now, we can simplify the fraction inside the parentheses. When we raise a fraction to a power, we raise the numerator and denominator to that power. Therefore, $\left(\frac{1}{y^2}\right)^4$ can be rewritten as $\frac{1}{y^8}$ .

${ \left(3x^3 \left(\frac{1}{y^2}\right)^4\right) = \left(3x^3 \frac{1}{y^8}\right) }$

Combining the Terms

Now, we can combine the terms inside the parentheses. When we multiply two terms with the same base, we add the exponents. Therefore, $x^3 \times x^0$ can be rewritten as $x^3$ .

${ \left(3x^3 \frac{1}{y^8}\right) = \frac{3x^3}{y^8} }$

Final Answer

Therefore, the simplified expression is $\frac{3x^3}{y^8}$ .

Q&A

Q: What is the rule for raising a power to a power?

A: When we raise a power to a power, we multiply the exponents.

Q: How do we simplify a fraction raised to a power?

A: When we raise a fraction to a power, we raise the numerator and denominator to that power.

Q: How do we combine terms with the same base?

A: When we multiply two terms with the same base, we add the exponents.

Q: What is the final answer to the expression?

A: The final answer is $\frac{3x^3}{y^8}$ .

Conclusion

Simplifying expressions is an essential skill in mathematics. By applying the rules of exponents and algebra, we can simplify complex expressions and make them easier to work with. In this article, we simplified the expression ${ \left(3x^3 \left(y^{-2}\right)4\right) }$ and arrived at the final answer of $\frac{3x^3}{y^8}$ . We also answered some common questions related to simplifying expressions.