Use The Respective Power Rule To Simplify The Expression:19) 4 Y − 7 X − 5 4y^{-7}x^{-5} 4 Y − 7 X − 5

Mar 12, 2025 by ADMIN 103 views

**Simplifying Expressions Using the Power Rule**

Introduction to the Power Rule

The power rule is a fundamental concept in algebra that allows us to simplify expressions involving exponents. It states that when we multiply two powers with the same base, we can add their exponents. In this article, we will focus on simplifying the expression $4y^{-7}x^{-5}$ using the power rule.

Understanding the Power Rule

The power rule is a simple yet powerful tool that helps us simplify complex expressions. It is based on the concept of exponents, which are numbers that represent repeated multiplication. For example, $x^3$ means $x$ multiplied by itself three times, or $x \times x \times x$ . When we multiply two powers with the same base, we can add their exponents. For instance, $x^3 \times x^2 = x^{3+2} = x^5$ .

Applying the Power Rule to the Expression

Now that we have a good understanding of the power rule, let's apply it to the expression $4y^{-7}x^{-5}$ . To simplify this expression, we need to multiply the coefficients and add the exponents of the variables. The coefficient of the expression is 4, which is a constant and does not affect the exponent. The variables are $y$ and $x$ , which have exponents of -7 and -5, respectively.

Simplifying the Expression

Using the power rule, we can simplify the expression as follows:

$4y^{-7}x^{-5} = 4 \times (y^{-7} \times x^{-5})$

$= 4 \times y^{-7-5}$

$= 4 \times y^{-12}$

$= \frac{4}{y^{12}}$

Conclusion

In this article, we have learned how to simplify the expression $4y^{-7}x^{-5}$ using the power rule. We have applied the power rule to multiply the coefficients and add the exponents of the variables. The resulting simplified expression is $\frac{4}{y^{12}}$ . This is a fundamental concept in algebra that helps us simplify complex expressions and solve problems involving exponents.

Examples and Practice Problems

Here are some examples and practice problems to help you practice simplifying expressions using the power rule:

Simplify the expression $3x^4y^{-2}$ using the power rule.
Simplify the expression $2y^3x^{-5}$ using the power rule.
Simplify the expression $5x^2y^{-4}$ using the power rule.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions using the power rule:

Make sure to multiply the coefficients and add the exponents of the variables.
Use the power rule to simplify complex expressions involving exponents.
Practice, practice, practice! The more you practice, the more comfortable you will become with simplifying expressions using the power rule.

Real-World Applications

The power rule has many real-world applications in fields such as science, engineering, and economics. For example, in physics, the power rule is used to calculate the energy of a system. In engineering, the power rule is used to design and optimize systems. In economics, the power rule is used to model and analyze economic systems.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying expressions using the power rule:

Not multiplying the coefficients and adding the exponents of the variables.
Not using the power rule to simplify complex expressions involving exponents.
Not practicing, practicing, practicing! The more you practice, the more comfortable you will become with simplifying expressions using the power rule.

Conclusion

In conclusion, the power rule is a fundamental concept in algebra that helps us simplify expressions involving exponents. By applying the power rule, we can simplify complex expressions and solve problems involving exponents. With practice and patience, you will become proficient in simplifying expressions using the power rule.

Q: What is the power rule in algebra?

A: The power rule is a fundamental concept in algebra that allows us to simplify expressions involving exponents. It states that when we multiply two powers with the same base, we can add their exponents.

Q: How do I apply the power rule to simplify an expression?

A: To apply the power rule, you need to multiply the coefficients and add the exponents of the variables. For example, if you have the expression $3x^4y^{-2}$ , you would multiply the coefficient 3 by the product of the variables $x^4$ and $y^{-2}$ , and then add the exponents of the variables.

Q: What is the difference between a coefficient and a variable?

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value. For example, in the expression $3x^4y^{-2}$ , the coefficient is 3 and the variables are $x$ and $y$ .

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent using the power rule. For example, if you have the expression $y^{-3}$ , you can simplify it by multiplying the coefficient 1 by the product of the variable $y$ and $y^{-3}$ , and then adding the exponents of the variable.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to multiply the coefficients and add the exponents of each variable. For example, if you have the expression $3x^4y^{-2}z^3$ , you would multiply the coefficient 3 by the product of the variables $x^4$ , $y^{-2}$ , and $z^3$ , and then add the exponents of each variable.

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent using the power rule. For example, if you have the expression $x^0$ , you can simplify it by multiplying the coefficient 1 by the product of the variable $x$ and $x^0$ , and then adding the exponents of the variable.

Q: How do I simplify an expression with a negative coefficient?

A: To simplify an expression with a negative coefficient, you need to multiply the coefficient by the product of the variables and then add the exponents of the variables. For example, if you have the expression $-3x^4y^{-2}$ , you would multiply the coefficient -3 by the product of the variables $x^4$ and $y^{-2}$ , and then add the exponents of the variables.

Q: Can I simplify an expression with a fractional exponent?

A: Yes, you can simplify an expression with a fractional exponent using the power rule. For example, if you have the expression $x^{1/2}$ , you can simplify it by multiplying the coefficient 1 by the product of the variable $x$ and $x^{1/2}$ , and then adding the exponents of the variable.

Q: How do I simplify an expression with a variable in the denominator?

A: To simplify an expression with a variable in the denominator, you need to multiply the coefficient by the product of the variables and then add the exponents of the variables. For example, if you have the expression $y^{-2}/x^3$ , you would multiply the coefficient 1 by the product of the variables $y^{-2}$ and $x^{-3}$ , and then add the exponents of the variables.

Q: Can I simplify an expression with a complex number?

A: Yes, you can simplify an expression with a complex number using the power rule. For example, if you have the expression $3x^4y^{-2}i$ , you would multiply the coefficient 3 by the product of the variables $x^4$ and $y^{-2}$ , and then add the exponents of the variables.

Q: How do I simplify an expression with a variable in the numerator and denominator?

A: To simplify an expression with a variable in the numerator and denominator, you need to multiply the coefficient by the product of the variables and then add the exponents of the variables. For example, if you have the expression $y^{-2}/x^3$ , you would multiply the coefficient 1 by the product of the variables $y^{-2}$ and $x^{-3}$ , and then add the exponents of the variables.