Simplify The Expression: ${ \left(-3 A^4 B\right)\left(4 A B^6\right) + \left(a^3 B^5\right)\left(-7 A^2 B^2\right) }$

Introduction

In algebra, simplifying expressions is a crucial skill that helps in solving complex equations and problems. It involves combining like terms, applying the rules of exponents, and rearranging the terms to make the expression more manageable. In this article, we will focus on simplifying the given expression: ${ \left(-3 a^4 b\right)\left(4 a b^6\right) + \left(a^3 b^5\right)\left(-7 a^2 b^2\right) }$

Understanding the Expression

The given expression consists of two terms, each of which is a product of two binomials. The first term is $\left(-3 a^4 b\right)\left(4 a b^6\right)$, and the second term is $\left(a^3 b^5\right)\left(-7 a^2 b^2\right)$. To simplify the expression, we need to apply the rules of exponents and combine like terms.

Applying the Rules of Exponents

When multiplying two terms with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$. Similarly, when multiplying two terms with the same base and different exponents, we multiply the coefficients and add the exponents. For example, $a^m \cdot b^n = a^m \cdot b^n$.

Simplifying the First Term

To simplify the first term, we need to multiply the coefficients and add the exponents. The coefficient of the first term is -3, and the coefficient of the second term is 4. The exponents of the first term are $a^4$ and $b^6$, and the exponents of the second term are $a^1$ and $b^6$. Therefore, the simplified form of the first term is:

$$\left(-3 a^4 b\right)\left(4 a b^6\right) = -12 a^5 b^7$$

Simplifying the Second Term

To simplify the second term, we need to multiply the coefficients and add the exponents. The coefficient of the first term is $a^3$, and the coefficient of the second term is $-7 a^2$. The exponents of the first term are $b^5$, and the exponents of the second term are $b^2$. Therefore, the simplified form of the second term is:

$$\left(a^3 b^5\right)\left(-7 a^2 b^2\right) = -7 a^5 b^7$$

Combining Like Terms

Now that we have simplified both terms, we can combine like terms. The two terms have the same exponent, $a^5 b^7$, but different coefficients. Therefore, we can combine them by adding the coefficients:

$$-12 a^5 b^7 + (-7 a^5 b^7) = (-12 - 7) a^5 b^7 = -19 a^5 b^7$$

Conclusion

In conclusion, the simplified form of the given expression is $\boxed{-19 a^5 b^7}$. This involves applying the rules of exponents, combining like terms, and rearranging the terms to make the expression more manageable. By following these steps, we can simplify complex expressions and solve problems in algebra.

Frequently Asked Questions

• Q: What is the rule for multiplying two terms with the same base? A: When multiplying two terms with the same base, we add the exponents.
• Q: What is the rule for multiplying two terms with the same base and different exponents? A: When multiplying two terms with the same base and different exponents, we multiply the coefficients and add the exponents.
• Q: How do we combine like terms? A: We combine like terms by adding the coefficients.

Final Thoughts

Simplifying expressions is a crucial skill in algebra that helps in solving complex equations and problems. By applying the rules of exponents, combining like terms, and rearranging the terms, we can simplify complex expressions and solve problems in algebra. In this article, we have simplified the given expression and provided a step-by-step guide on how to simplify expressions. We hope that this article has provided valuable insights and helped in understanding the concept of simplifying expressions in algebra.

Introduction

In our previous article, we discussed how to simplify the expression: ${ \left(-3 a^4 b\right)\left(4 a b^6\right) + \left(a^3 b^5\right)\left(-7 a^2 b^2\right) }$. We applied the rules of exponents, combined like terms, and rearranged the terms to make the expression more manageable. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights on simplifying expressions.

Q&A Section

Q: What is the rule for multiplying two terms with the same base?

A: When multiplying two terms with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the rule for multiplying two terms with the same base and different exponents?

A: When multiplying two terms with the same base and different exponents, we multiply the coefficients and add the exponents. For example, $a^m \cdot b^n = a^m \cdot b^n$.

Q: How do we combine like terms?

A: We combine like terms by adding the coefficients. For example, $a^m + a^n = a^m + a^n$.

Q: What is the difference between combining like terms and simplifying expressions?

A: Combining like terms involves adding or subtracting terms with the same exponent, while simplifying expressions involves rearranging the terms to make the expression more manageable.

Q: Can we simplify expressions with negative exponents?

A: Yes, we can simplify expressions with negative exponents by applying the rule $a^{-m} = \frac{1}{a^m}$.

Q: How do we simplify expressions with fractional exponents?

A: We can simplify expressions with fractional exponents by applying the rule $a^{m/n} = \sqrt[n]{a^m}$.

Q: Can we simplify expressions with variables in the exponent?

A: Yes, we can simplify expressions with variables in the exponent by applying the rule $a^{m+n} = a^m \cdot a^n$.

Q: How do we simplify expressions with multiple terms?

A: We can simplify expressions with multiple terms by applying the rules of exponents, combining like terms, and rearranging the terms.

Q: Can we simplify expressions with absolute values?

A: Yes, we can simplify expressions with absolute values by applying the rule $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$.

Conclusion

In conclusion, simplifying expressions is a crucial skill in algebra that helps in solving complex equations and problems. By applying the rules of exponents, combining like terms, and rearranging the terms, we can simplify complex expressions and solve problems in algebra. We hope that this Q&A section has provided valuable insights and helped in understanding the concept of simplifying expressions in algebra.

Frequently Asked Questions

• Q: What is the rule for multiplying two terms with the same base? A: When multiplying two terms with the same base, we add the exponents.
• Q: What is the rule for multiplying two terms with the same base and different exponents? A: When multiplying two terms with the same base and different exponents, we multiply the coefficients and add the exponents.
• Q: How do we combine like terms? A: We combine like terms by adding the coefficients.

Final Thoughts

Simplifying expressions is a crucial skill in algebra that helps in solving complex equations and problems. By applying the rules of exponents, combining like terms, and rearranging the terms, we can simplify complex expressions and solve problems in algebra. We hope that this Q&A section has provided valuable insights and helped in understanding the concept of simplifying expressions in algebra.