Problem 78 - Elementary Algebra

Question

Assuming the bases are not zero, find the value of \(\left(4 a^{2} b^{3}\right)\left(5 a b^{4}\right)\).

Step-by-Step Solution

Answer

Answer: The value of (𝟒𝒂²𝒃³)(𝟓𝒂𝒃⁴) is 20𝒂³𝒃⁷.

Step 1: Multiply the constants

Start by multiplying the constants \(4\) and \(5\). This gives \(4\times 5=20\).

Step 2: Apply the product rule for the exponent of \(a\)

Next, we need to apply the product rule for the exponent of \(a\). The product rule states that, when multiplying two quantities with the same base, we can add the exponents. Here we have \(a^2\times a^1\) (note that \(a\) can also be written as \(a^1\)). So, we will add the exponents: \(2 + 1 = 3\). The result is \(a^3\).

Step 3: Apply the product rule for the exponent of \(b\)

Now, we will do the same thing for the exponent of \(b\). We have \(b^3\times b^4\). Adding the exponents: \(3+4=7\). The result is \(b^7\).

Step 4: Combine the results

Combine the results from steps 1 to 3 to get the final expression: \(20\,a^3b^7\). Therefore, the value of \(\left(4 a^{2}b^{3}\right)\left(5 a b^{4}\right)\) is \(20\,a^3b^7\).

Key Concepts

Exponentiation in Algebra

Exponentiation is a fundamental concept in algebra, where numbers are raised to a certain power, indicating the number of times a base is multiplied by itself.

For example, the expression \(a^n\) tells us to multiply \(a\), the base, by itself \(n\) times, where \(n\) is the exponent. In the context of the original exercise, we have \(4 a^{2} b^{3}\) and \(5 a b^{4}\), which when expanded, would look like \(4 \cdot a \cdot a \cdot b \cdot b \cdot b\) and \(5 \cdot a \cdot b \cdot b \cdot b \cdot b\).

Instead of expanding these expressions and multiplying them term by term, algebra offers us properties and rules that significantly simplify these calculations. As shown in the step-by-step solution, by using the product rule for exponents, we can find the product much more efficiently.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves the reduction of these expressions into their most compact and simple form, using arithmetic operations and algebraic properties.

The purpose is to make the expressions easier to understand and work with. Applying properties like the distributive property, the commutative property of multiplication, and especially the properties of exponents, we can transform expressions without altering their values.

Steps to Simplify

• Multiply the coefficients, just like the numbers 4 and 5 in our expression, which results in 20.
• For variables, use the product rule: add exponents of like bases.

So, in the exercise, we merged the expressions \(a^2 \cdot a^1\) to \(a^3\) and \(b^3 \cdot b^4\) to \(b^7\). Lastly, the components are combined to yield a simplified form: \(20a^3b^7\).

Properties of Exponents

Properties of exponents offer a set of rules that make working with exponential expressions more manageable.

The product rule is a primary focus in the given exercise. This rule states that when we multiply two expressions with the same base, we add their exponents. It’s the reason we get \(a^{2+1} = a^3\) and \(b^{3+4} = b^7\) when we multiply \(a^2 \cdot a\) and \(b^3 \cdot b^4\), respectively.

Other Important Properties

• The power rule: \(a^{m} \cdot a^{n} = a^{m+n}\)
• The quotient rule: \(\frac{a^{m}}{a^{n}} = a^{m-n}\)
• The power of a product rule: \(\left(ab\right)^{n} = a^{n} \cdot b^{n}\)
• The power of a power rule: \(\left(a^{m}\right)^{n} = a^{m \cdot n}\)

By understanding these properties, students are able to simplify complex expressions confidently and accurately, leading to a clearer grasp of algebra and its practical applications.