Problem 6
Question
In Exercises, use the properties of exponents to simplify the expression. (a) \(\left(4^{3}\right)\left(4^{2}\right)\) (b) \(\left(\frac{1}{4}\right)^{2}\left(4^{2}\right)\) (c) \(\left(4^{6}\right)^{1 / 2}\) (d) \(\left[\left(8^{-1}\right)\left(8^{2 / 3}\right)\right]^{3}\)
Step-by-Step Solution
Verified Answer
(a) \(4^{5}\) (b) \(1\) (c) \(4^{3}\) (d) \(8^{-1}\)
1Step 1: Simplify Expression (a)
For expression (a) \(\left(4^{3}\right)\left(4^{2}\right)\), apply the product of powers rule - When multiplying like bases, you can add the exponents. Thus, \(4^{3 + 2} = 4^{5}\)
2Step 2: Simplify Expression (b)
For expression (b) \(\left(\frac{1}{4}\right)^{2}\left(4^{2}\right)\), first simplify the power \(\left(\frac{1}{4}\right)^{2}\) to \(\frac{1}{16}\), then multiply \(\frac{1}{16}\) by \(4^2 = 16\) to get \(1\).
3Step 3: Simplify Expression (c)
For expression (c) \(\left(4^{6}\right)^{1 / 2}\), apply the power of a power rule - When raising a power to a power, multiply the exponents. Thus, \(4^{6*(1/2)} = 4^{3}\)
4Step 4: Simplify Expression (d)
For expression (d) \(\left[\left(8^{-1}\right)\left(8^{2 / 3}\right)\right]^{3}\), apply the power rule to combine the two terms inside the brackets \(\left(8^{(-1 + 2/3)}\right)^3\). Then, simplify \((-1 + 2/3)\) to \(-1/3\) giving \(\left(8^{-1 / 3}\right)^3\). Apply power of a power rule to get \(8^{-1}\).
Key Concepts
Product of Powers RulePower of a Power RuleSimplifying Algebraic Expressions
Product of Powers Rule
Understanding the product of powers rule is essential when working with exponents. This rule simply states that when you multiply two expressions with the same base, you can keep the base and add the exponents. It's an elegant shortcut that simplifies multiplication of exponential terms.
Let's look at an example directly from our exercise. Given the expression \(4^{3}\times4^{2}\), instead of multiplying \(4\) by itself five times, we use the product of powers rule. Here's how it works:
Let's look at an example directly from our exercise. Given the expression \(4^{3}\times4^{2}\), instead of multiplying \(4\) by itself five times, we use the product of powers rule. Here's how it works:
- Identify the common base, which is \(4\) in this instance.
- Add the exponents: \(3 + 2 = 5\).
- Write the simplified expression: \(4^5\).
Power of a Power Rule
The power of a power rule takes simplifying exponents a step further. It is used when you have an exponent raised to another exponent, like a power tower. According to this rule, you multiply the exponents together. This simplifies expressions where you might otherwise be stuck calculating a large power first, only to raise it to another power.
In our example \(\left(4^{6}\right)^{1/2}\), you don't have to calculate \(4^6\) first. Here's the process using the power of a power rule:
In our example \(\left(4^{6}\right)^{1/2}\), you don't have to calculate \(4^6\) first. Here's the process using the power of a power rule:
- Start with the base. The base of our inner exponent is \(4\).
- Multiply the exponents together. For us, it's \(6\times{1/2}=3\).
- Combine the base with the new exponent: \(4^3\).
Simplifying Algebraic Expressions
Simplifying algebraic expressions is a vital skill in algebra that involves reducing expressions to their simplest form by performing operations and applying rules like those for exponents. The goal is to make the expressions as concise and as clear as possible, often to make them easier to understand or to solve.
Using the properties of exponents, such as the product of powers and the power of a power rules, we can transform complicated-looking expressions into their simplest form. In our series of exercises, you've seen these rules in action. For instance, in expression (d) \(\left[\left(8^{-1}\right)\left(8^{2 / 3}\right)\right]^{3}\), we first use the product of powers rule inside the brackets, then apply the power of a power rule to the result.
This step-by-step approach allows us to work through each part of the expression methodically, ensuring that each operation is carried out correctly and the expression becomes simpler at each stage. It is crucial to follow the correct order of operations and to meticulously apply each relevant algebraic rule to achieve the desired simplification.
Using the properties of exponents, such as the product of powers and the power of a power rules, we can transform complicated-looking expressions into their simplest form. In our series of exercises, you've seen these rules in action. For instance, in expression (d) \(\left[\left(8^{-1}\right)\left(8^{2 / 3}\right)\right]^{3}\), we first use the product of powers rule inside the brackets, then apply the power of a power rule to the result.
This step-by-step approach allows us to work through each part of the expression methodically, ensuring that each operation is carried out correctly and the expression becomes simpler at each stage. It is crucial to follow the correct order of operations and to meticulously apply each relevant algebraic rule to achieve the desired simplification.
Other exercises in this chapter
Problem 6
In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ e^{2}=7.3891 \ldots $$
View solution Problem 6
In Exercises, find the derivative of the function. $$ y=e^{1-x} $$
View solution Problem 7
In Exercises, use the given information to write an equation for \(y\). Confirm your result analytically by showing that the function satisfies the equation \(d
View solution Problem 7
In Exercises, find the derivative of the function. $$ y=e^{-x^{2}} $$
View solution