Unit: Expanding PolynomialsSimplify The Expression: \left(2 A^3 B\right)^3\left(-2 B^4\right ]A. − 4 A 9 B 7 -4 A^9 B^7 − 4 A 9 B 7 B. − 16 A 9 B 7 -16 A^9 B^7 − 16 A 9 B 7 C. − 16 A 9 B 12 -16 A^9 B^{12} − 16 A 9 B 12 D. 16 A 9 B 7 16 A^9 B^7 16 A 9 B 7

Introduction

Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. Expanding polynomials involves simplifying complex expressions by applying the rules of exponents and algebraic manipulation. In this article, we will focus on simplifying the expression $\left(2 a^3 b\right)^3\left(-2 b^4\right)$ using the properties of exponents and algebraic manipulation.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $a^3$ represents $a \times a \times a$. When we have an expression with exponents, we can simplify it by applying the rules of exponents. The rules of exponents state that:

• When we multiply two powers with the same base, we add the exponents. For example, $a^3 \times a^2 = a^{3+2} = a^5$.
• When we divide two powers with the same base, we subtract the exponents. For example, $\frac{a^3}{a^2} = a^{3-2} = a^1$.
• When we raise a power to a power, we multiply the exponents. For example, $(a^3)^2 = a^{3 \times 2} = a^6$.

Simplifying the Expression

Now that we have a good understanding of exponents, let's simplify the expression $\left(2 a^3 b\right)^3\left(-2 b^4\right)$. To simplify this expression, we will apply the rules of exponents and algebraic manipulation.

First, let's expand the expression using the distributive property:

$\left(2 a^3 b\right)^3\left(-2 b^4\right) = \left(2 a^3 b\right)\left(2 a^3 b\right)\left(2 a^3 b\right)\left(-2 b^4\right)$

Next, let's simplify each term using the rules of exponents:

$\left(2 a^3 b\right)\left(2 a^3 b\right)\left(2 a^3 b\right)\left(-2 b^4\right) = 2^3 a^{3+3+3} b^{1+1+1}\left(-2 b^4\right)$

Now, let's simplify the exponents:

$2^3 a^{3+3+3} b^{1+1+1}\left(-2 b^4\right) = 8 a^9 b^3\left(-2 b^4\right)$

Next, let's simplify the expression by multiplying the coefficients:

$8 a^9 b^3\left(-2 b^4\right) = -16 a^9 b^7$

Conclusion

In this article, we simplified the expression $\left(2 a^3 b\right)^3\left(-2 b^4\right)$ using the properties of exponents and algebraic manipulation. We applied the rules of exponents to simplify the expression and arrived at the final answer: $-16 a^9 b^7$. This expression is a simplified form of the original expression, and it demonstrates the importance of understanding exponents and algebraic manipulation in simplifying complex expressions.

Discussion

The expression $\left(2 a^3 b\right)^3\left(-2 b^4\right)$ is a classic example of a polynomial expression that requires simplification using the rules of exponents and algebraic manipulation. The correct answer is $-16 a^9 b^7$, which demonstrates the importance of understanding exponents and algebraic manipulation in simplifying complex expressions.

Practice Problems

1. Simplify the expression $\left(3 a^2 b\right)^2\left(-3 b^3\right)$ using the properties of exponents and algebraic manipulation.
2. Simplify the expression $\left(4 a^4 b\right)^3\left(-4 b^2\right)$ using the properties of exponents and algebraic manipulation.
3. Simplify the expression $\left(2 a^3 b\right)^2\left(-2 b^5\right)$ using the properties of exponents and algebraic manipulation.

Answer Key

1. $-9 a^6 b^6$
2. $-64 a^{12} b^8$
3. $-4 a^6 b^7$

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Q: What is the difference between expanding and simplifying polynomials?

A: Expanding polynomials involves breaking down a complex expression into simpler terms, while simplifying polynomials involves reducing a complex expression to its simplest form. In the case of the expression $\left(2 a^3 b\right)^3\left(-2 b^4\right)$, we expanded the expression using the distributive property and then simplified it using the rules of exponents.

Q: How do I know when to use the distributive property and when to use the rules of exponents?

A: The distributive property is used when we have a product of two or more terms, and we want to expand it into a sum of terms. The rules of exponents are used when we have an expression with exponents, and we want to simplify it. In the case of the expression $\left(2 a^3 b\right)^3\left(-2 b^4\right)$, we used the distributive property to expand the expression and then used the rules of exponents to simplify it.

Q: What is the order of operations when simplifying polynomials?

A: The order of operations when simplifying polynomials is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate expressions with exponents next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate addition and subtraction operations from left to right.

Q: How do I simplify expressions with negative exponents?

A: When simplifying expressions with negative exponents, we can use the rule that $a^{-n} = \frac{1}{a^n}$. For example, if we have the expression $a^{-3}$, we can simplify it by writing it as $\frac{1}{a^3}$.

Q: Can I simplify expressions with fractional exponents?

A: Yes, we can simplify expressions with fractional exponents using the rule that $a^{m/n} = \sqrt[n]{a^m}$. For example, if we have the expression $a^{3/2}$, we can simplify it by writing it as $\sqrt{a^3}$.

Q: How do I simplify expressions with multiple variables?

A: When simplifying expressions with multiple variables, we can use the rules of exponents to simplify each variable separately. For example, if we have the expression $(2 a^3 b)^3 (-2 b^4)$, we can simplify it by first simplifying the expression inside the parentheses and then simplifying the expression outside the parentheses.

Q: Can I simplify expressions with radicals?

A: Yes, we can simplify expressions with radicals using the rule that $\sqrt[n]{a^m} = a^{m/n}$. For example, if we have the expression $\sqrt{a^3}$, we can simplify it by writing it as $a^{3/2}$.

Q: How do I know when to use the conjugate to simplify expressions with radicals?

A: The conjugate is used to simplify expressions with radicals when the expression is in the form $a + \sqrt{b}$. The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$. We can multiply the expression by the conjugate to eliminate the radical.

Q: Can I simplify expressions with absolute values?

A: Yes, we can simplify expressions with absolute values using the rule that $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$. For example, if we have the expression $|a^2 - 4|$, we can simplify it by writing it as $a^2 - 4$ if $a^2 - 4 \geq 0$ and $-(a^2 - 4)$ if $a^2 - 4 < 0$.

Q: How do I know when to use the quadratic formula to simplify expressions?

A: The quadratic formula is used to simplify expressions in the form $ax^2 + bx + c = 0$. We can use the quadratic formula to find the solutions to the equation.

Q: Can I simplify expressions with trigonometric functions?

A: Yes, we can simplify expressions with trigonometric functions using the rules of trigonometry. For example, if we have the expression $\sin^2 x + \cos^2 x$, we can simplify it by writing it as $1$.

Q: How do I know when to use the Pythagorean identity to simplify expressions?

A: The Pythagorean identity is used to simplify expressions in the form $\sin^2 x + \cos^2 x$. We can use the Pythagorean identity to simplify the expression.

Q: Can I simplify expressions with logarithmic functions?

A: Yes, we can simplify expressions with logarithmic functions using the rules of logarithms. For example, if we have the expression $\log_a (b^c)$, we can simplify it by writing it as $c \log_a b$.

Q: How do I know when to use the change of base formula to simplify expressions?

A: The change of base formula is used to simplify expressions in the form $\log_a b$. We can use the change of base formula to simplify the expression.

Q: Can I simplify expressions with exponential functions?

A: Yes, we can simplify expressions with exponential functions using the rules of exponents. For example, if we have the expression $a^{b+c}$, we can simplify it by writing it as $a^b \cdot a^c$.

Q: How do I know when to use the product rule to simplify expressions?

A: The product rule is used to simplify expressions in the form $(a \cdot b)^c$. We can use the product rule to simplify the expression.

Q: Can I simplify expressions with rational expressions?

A: Yes, we can simplify expressions with rational expressions using the rules of fractions. For example, if we have the expression $\frac{a}{b} + \frac{c}{d}$, we can simplify it by finding a common denominator.

Q: How do I know when to use the least common multiple to simplify expressions?

A: The least common multiple is used to simplify expressions in the form $\frac{a}{b} + \frac{c}{d}$. We can use the least common multiple to simplify the expression.

Q: Can I simplify expressions with complex numbers?

A: Yes, we can simplify expressions with complex numbers using the rules of complex numbers. For example, if we have the expression $a + bi$, we can simplify it by writing it as $a + bi$.

Q: How do I know when to use the conjugate to simplify expressions with complex numbers?

A: The conjugate is used to simplify expressions with complex numbers when the expression is in the form $a + bi$. We can use the conjugate to simplify the expression.

Q: Can I simplify expressions with matrices?

A: Yes, we can simplify expressions with matrices using the rules of matrices. For example, if we have the expression $A + B$, we can simplify it by adding the corresponding elements of the matrices.

Q: How do I know when to use the determinant to simplify expressions with matrices?

A: The determinant is used to simplify expressions with matrices when the expression is in the form $A \cdot B$. We can use the determinant to simplify the expression.

Q: Can I simplify expressions with vectors?

A: Yes, we can simplify expressions with vectors using the rules of vectors. For example, if we have the expression $A + B$, we can simplify it by adding the corresponding elements of the vectors.

Q: How do I know when to use the dot product to simplify expressions with vectors?

A: The dot product is used to simplify expressions with vectors when the expression is in the form $A \cdot B$. We can use the dot product to simplify the expression.

Q: Can I simplify expressions with differential equations?

A: Yes, we can simplify expressions with differential equations using the rules of differential equations. For example, if we have the expression $\frac{dy}{dx} = f(x)$, we can simplify it by finding the solution to the differential equation.

Q: How do I know when to use the separation of variables to simplify expressions with differential equations?

A: The separation of variables is used to simplify expressions with differential equations when the expression is in the form $\frac{dy}{dx} = f(x)$. We can use the separation of variables to simplify the expression.

Q: Can I simplify expressions with integral equations?

A: Yes, we can simplify expressions with integral equations using the rules of integral equations. For example, if we have the expression $\int_{a}^{b} f(x) dx$, we can simplify it by finding the antiderivative of the function.

Q: How do I know when to use the fundamental theorem of calculus to simplify expressions with integral equations?

A: The fundamental theorem of calculus is used to simplify expressions with integral equations when the expression is in the form $\int_{a}^{b} f(x) dx$. We can use the fundamental theorem of calculus to simplify the expression.

**Q: Can I simplify expressions with differential