Find The Product Of $7 A^4 B$ And $\left(-3 A^5 B^6\right)^2$.

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Understanding the Problem

In this problem, we are tasked with finding the product of two algebraic expressions: $7 a^4 b$ and $\left(-3 a^5 b6\right)2$. To solve this problem, we need to apply the rules of exponents and multiplication of algebraic expressions.

Applying the Rules of Exponents

The first step in solving this problem is to apply the rules of exponents. Specifically, we need to use the power of a power rule, which states that for any numbers aa and bb and any integers mm and nn, we have:

(am)n=amâ‹…n\left(a^m\right)^n = a^{m \cdot n}

Using this rule, we can rewrite the second expression as:

(−3a5b6)2=(−3)2⋅(a5)2⋅(b6)2\left(-3 a^5 b^6\right)^2 = (-3)^2 \cdot \left(a^5\right)^2 \cdot \left(b^6\right)^2

Simplifying the Expression

Now that we have rewritten the second expression, we can simplify it further by applying the power of a power rule:

(−3)2⋅(a5)2⋅(b6)2=9⋅a5⋅2⋅b6⋅2(-3)^2 \cdot \left(a^5\right)^2 \cdot \left(b^6\right)^2 = 9 \cdot a^{5 \cdot 2} \cdot b^{6 \cdot 2}

=9â‹…a10â‹…b12= 9 \cdot a^{10} \cdot b^{12}

Multiplying the Two Expressions

Now that we have simplified the second expression, we can multiply it with the first expression:

7a4bâ‹…9a10b127 a^4 b \cdot 9 a^{10} b^{12}

Applying the Product of Powers Rule

To multiply the two expressions, we need to apply the product of powers rule, which states that for any numbers aa and bb and any integers mm and nn, we have:

amâ‹…an=am+na^m \cdot a^n = a^{m + n}

Using this rule, we can rewrite the product as:

7a4bâ‹…9a10b12=7â‹…9â‹…a4+10â‹…b1+127 a^4 b \cdot 9 a^{10} b^{12} = 7 \cdot 9 \cdot a^{4 + 10} \cdot b^{1 + 12}

Simplifying the Product

Now that we have rewritten the product, we can simplify it further by applying the product of powers rule:

7â‹…9â‹…a4+10â‹…b1+12=63â‹…a14â‹…b137 \cdot 9 \cdot a^{4 + 10} \cdot b^{1 + 12} = 63 \cdot a^{14} \cdot b^{13}

The Final Answer

Therefore, the product of $7 a^4 b$ and $\left(-3 a^5 b6\right)2$ is:

63a14b1363 a^{14} b^{13}

Conclusion

In this problem, we applied the rules of exponents and multiplication of algebraic expressions to find the product of two algebraic expressions. We used the power of a power rule, the product of powers rule, and simplified the expressions to arrive at the final answer.

Frequently Asked Questions

Q: What is the product of $7 a^4 b$ and $\left(-3 a^5 b6\right)2$?

A: The product is $63 a^{14} b^{13}$.

Q: How do I apply the rules of exponents to solve this problem?

A: To solve this problem, you need to apply the power of a power rule and the product of powers rule.

Q: What is the power of a power rule?

A: The power of a power rule states that for any numbers aa and bb and any integers mm and nn, we have:

(am)n=amâ‹…n\left(a^m\right)^n = a^{m \cdot n}

Q: What is the product of powers rule?

A: The product of powers rule states that for any numbers aa and bb and any integers mm and nn, we have:

amâ‹…an=am+na^m \cdot a^n = a^{m + n}

References

  • [1] Algebraic Expressions, Wikipedia
  • [2] Rules of Exponents, Math Open Reference
  • [3] Multiplication of Algebraic Expressions, Purplemath

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Understanding Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics, and they play a crucial role in solving equations and inequalities. In this article, we will explore the world of algebraic expressions, including their definition, types, and rules for simplifying and manipulating them.

Frequently Asked Questions

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: What are the different types of algebraic expressions?

A: There are several types of algebraic expressions, including:

  • Polynomial expressions: These are algebraic expressions that consist of variables and constants, and the variables are raised to non-negative integer powers.
  • Rational expressions: These are algebraic expressions that consist of variables and constants, and the variables are raised to non-negative integer powers, and the expression is divided by a non-zero constant.
  • Algebraic fractions: These are algebraic expressions that consist of variables and constants, and the variables are raised to non-negative integer powers, and the expression is divided by a non-zero constant.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to apply the rules of exponents, combine like terms, and eliminate any unnecessary parentheses.

Q: What is the order of operations in algebraic expressions?

A: The order of operations in algebraic expressions is:

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

Q: How do I multiply algebraic expressions?

A: To multiply algebraic expressions, you need to apply the product of powers rule, which states that for any numbers aa and bb and any integers mm and nn, we have:

amâ‹…an=am+na^m \cdot a^n = a^{m + n}

Q: How do I divide algebraic expressions?

A: To divide algebraic expressions, you need to apply the quotient of powers rule, which states that for any numbers aa and bb and any integers mm and nn, we have:

am÷an=am−na^m \div a^n = a^{m - n}

Q: What is the difference between an algebraic expression and an equation?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations, while an equation is a statement that two algebraic expressions are equal.

Examples of Algebraic Expressions

Example 1: Simplifying an Algebraic Expression

Simplify the algebraic expression: $2x^2 + 3x - 4$

Solution:

2x2+3x−4=2x2+3x−42x^2 + 3x - 4 = 2x^2 + 3x - 4

Example 2: Multiplying Algebraic Expressions

Multiply the algebraic expressions: $x^2 + 2x - 3$ and $x + 1$

Solution:

(x2+2x−3)⋅(x+1)=x3+2x2−3x+x2+2x−3(x^2 + 2x - 3) \cdot (x + 1) = x^3 + 2x^2 - 3x + x^2 + 2x - 3

=x3+3x2−x−3= x^3 + 3x^2 - x - 3

Example 3: Dividing Algebraic Expressions

Divide the algebraic expressions: $x^2 + 2x - 3$ and $x + 1$

Solution:

(x2+2x−3)÷(x+1)=x−3(x^2 + 2x - 3) \div (x + 1) = x - 3

Conclusion

In this article, we have explored the world of algebraic expressions, including their definition, types, and rules for simplifying and manipulating them. We have also provided examples of algebraic expressions and demonstrated how to simplify, multiply, and divide them.

Frequently Asked Questions (FAQs)

Q: What is the difference between an algebraic expression and a numerical expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations, while a numerical expression is a mathematical expression that consists of only numbers and mathematical operations.

Q: How do I evaluate an algebraic expression?

A: To evaluate an algebraic expression, you need to substitute the values of the variables and constants into the expression and then simplify it.

Q: What is the difference between an algebraic expression and a function?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations, while a function is a relation between a set of inputs and a set of possible outputs.

References

  • [1] Algebraic Expressions, Wikipedia
  • [2] Rules of Exponents, Math Open Reference
  • [3] Multiplication and Division of Algebraic Expressions, Purplemath