Problem 69
Question
Evaluate the expression. \(\left(4^{5}\right)^{0}\)
Step-by-Step Solution
Verified Answer
The evaluated expression is 1.
1Step 1: Identify the base and the exponent
The base is \(4^{5}\) and the exponent is 0.
2Step 2: Apply the rule of exponentials
Any number to the power of 0 is 1, no matter what the base number is. Therefore, \(\left(4^{5}\right)^{0} = 1\)
Key Concepts
Properties of ExponentsEvaluating ExpressionsExponential Notation
Properties of Exponents
Like the shortcuts that make arithmetic operations easier, exponents in algebra come with their own set of rules or properties that simplify expressions and make solving equations more efficient. Understanding these properties is vital for any student of algebra.
The most basic properties include the product of powers which states that when you multiply same bases, you simply add the exponents, like in the expression \( x^a \cdot x^b = x^{a+b} \). Conversely, the quotient of powers property holds that dividing same bases equates to subtracting the exponents, \( x^a / x^b = x^{a-b} \).
Other important properties include the power of a power, where you multiply exponents when a base raised to an exponent is itself raised to another exponent (\((x^a)^b = x^{a \cdot b}\)). The power of a product property tells us that an exponent applies to all terms inside a parenthesis (\((xy)^a = x^a \cdot y^a\)). Lastly, any base raised to the exponent zero is equal to one, which was applied in our example problem (\(x^0 = 1\), regardless of the value of 'x').
These rules form the foundation for dealing with more complex algebraic operations involving exponents.
The most basic properties include the product of powers which states that when you multiply same bases, you simply add the exponents, like in the expression \( x^a \cdot x^b = x^{a+b} \). Conversely, the quotient of powers property holds that dividing same bases equates to subtracting the exponents, \( x^a / x^b = x^{a-b} \).
Other important properties include the power of a power, where you multiply exponents when a base raised to an exponent is itself raised to another exponent (\((x^a)^b = x^{a \cdot b}\)). The power of a product property tells us that an exponent applies to all terms inside a parenthesis (\((xy)^a = x^a \cdot y^a\)). Lastly, any base raised to the exponent zero is equal to one, which was applied in our example problem (\(x^0 = 1\), regardless of the value of 'x').
These rules form the foundation for dealing with more complex algebraic operations involving exponents.
Evaluating Expressions
Evaluating expressions is a fundamental process in algebra, which involves substituting values for variables and performing operations in accordance with the rules of arithmetic and properties of exponents.
To successfully evaluate expressions, start by simplifying the parts of the expression in parentheses followed by exponents, and then progress with multiplication and division from left to right, finishing with addition and subtraction (remembered easily by the acronym PEMDAS or BODMAS).
For the given problem \( (4^5)^0 \)), we first observe the exponent outside which is zero. According to the exponent properties, any expression raised to the zero power is 1, regardless of the complexity inside the parentheses. Hence, without needing to calculate \( 4^5 \) or any other operations, we can directly conclude \( (4^5)^0 = 1 \). This is an example of how the properties of exponents facilitate easier evaluation of expressions.
To successfully evaluate expressions, start by simplifying the parts of the expression in parentheses followed by exponents, and then progress with multiplication and division from left to right, finishing with addition and subtraction (remembered easily by the acronym PEMDAS or BODMAS).
For the given problem \( (4^5)^0 \)), we first observe the exponent outside which is zero. According to the exponent properties, any expression raised to the zero power is 1, regardless of the complexity inside the parentheses. Hence, without needing to calculate \( 4^5 \) or any other operations, we can directly conclude \( (4^5)^0 = 1 \). This is an example of how the properties of exponents facilitate easier evaluation of expressions.
Exponential Notation
Exponential notation is a concise way to represent repeated multiplication of the same factor. It's a notation that uses a base and an exponent where the base is the number being multiplied and the exponent represents how many times it is multiplied by itself. The expression \( b^n \) means 'b' multiplied by itself 'n' times.
For instance, \( 4^2 \) (read as 'four squared') means \( 4 \times 4 \), and \( 4^3 \) (read as 'four cubed') means \( 4 \times 4 \times 4 \). It's a powerful tool in algebra because it allows us to write and manipulate very large or very small numbers efficiently.
Exponential notation also aids in understanding the concept of exponential growth, common in science, finance, and statistics, where quantities double or halve over consistent intervals, represented by expressions like \( 2^n \) or \(\frac{1}{2}^n\).
For instance, \( 4^2 \) (read as 'four squared') means \( 4 \times 4 \), and \( 4^3 \) (read as 'four cubed') means \( 4 \times 4 \times 4 \). It's a powerful tool in algebra because it allows us to write and manipulate very large or very small numbers efficiently.
Exponential notation also aids in understanding the concept of exponential growth, common in science, finance, and statistics, where quantities double or halve over consistent intervals, represented by expressions like \( 2^n \) or \(\frac{1}{2}^n\).
Other exercises in this chapter
Problem 69
Write the equation in standard form. $$ 6 x^{2}=5 x-7 $$
View solution Problem 69
Add or subtract the polynomials. (Lesson 10.1) $$\left(-5 x^{2}+2 x-12\right)-\left(6-9 x-7 x^{2}\right)$$
View solution Problem 69
Find the sum. $$75.6+35.8$$
View solution Problem 70
Write the equation in standard form. $$ 9-6 x=2 x^{2} $$
View solution