Problem 32
Question
In the following exercises, evaluate the following expressions. When \(x=3\), (a) \(x^{5}\) (b) \(5^{x}\) (c) \(3 x^{2}-4 x-8\)
Step-by-Step Solution
Verified Answer
(a) 243, (b) 125, (c) 7
1Step 1: Substitute the value of x in part (a)
Given the expression is \(x^{5}\). Substitute \(x = 3\) into the expression. This gives \(3^{5}\).
2Step 2: Calculate the power
Calculate \(3^{5}\). This is \(3 \times 3 \times 3 \times 3 \times 3 = 243\). So, \(x^{5} = 243\) when \(x = 3\).
3Step 3: Substitute the value of x in part (b)
Given the expression is \(5^{x}\). Substitute \(x = 3\) into the expression. This gives \(5^{3}\).
4Step 4: Calculate the power
Calculate \(5^{3}\). This is \(5 \times 5 \times 5 = 125\). So, \(5^{x} = 125\) when \(x = 3\).
5Step 5: Substitute the value of x in part (c)
Given the expression is \(3 x^{2} - 4 x - 8\). Substitute \(x = 3\) into the expression. This gives \(3(3)^{2} - 4(3) - 8\).
6Step 6: Calculate the value
Calculate each term step by step. First, \(3(3)^{2} = 3 \times 9 = 27\). Next, \(4(3) = 12\). The expression now is \(27 - 12 - 8\). Simplifying this further gives \(27 - 12 - 8 = 7\). So, \(3x^{2} - 4x - 8 = 7\) when \(x = 3\).
Key Concepts
substitution in algebraexponentiationpolynomial evaluation
substitution in algebra
Substitution in algebra is a fundamental concept that involves replacing a variable with a given number to evaluate expressions.
This process simplifies expressions and helps in finding specific values for different scenarios.
For example, if we are given an expression like \(x^5\) and told to find its value when \(x=3\), we substitute the 3 into the expression.
It becomes \((3)^5\), which we then solve using exponentiation.
Substitution is straightforward but requires careful attention to ensure the correct value is inserted for each variable.
This process simplifies expressions and helps in finding specific values for different scenarios.
For example, if we are given an expression like \(x^5\) and told to find its value when \(x=3\), we substitute the 3 into the expression.
It becomes \((3)^5\), which we then solve using exponentiation.
Substitution is straightforward but requires careful attention to ensure the correct value is inserted for each variable.
exponentiation
Exponentiation is a mathematical operation involving a base and an exponent.
The base is the number being multiplied, while the exponent indicates how many times the base is used as a factor.
For example, in the given exercise, expressions like \(x^5\) or \(5^x\) involve exponentiation.
If we take \(x = 3\), then solving \(3^5\) by multiplying three by itself five times, we get \(243\).
Similarly, \(5^3\) means multiplying five by itself three times, equaling \(125\).
When dealing with exponentiation, it is essential to perform the multiplication accurately to get the correct result.
The base is the number being multiplied, while the exponent indicates how many times the base is used as a factor.
For example, in the given exercise, expressions like \(x^5\) or \(5^x\) involve exponentiation.
If we take \(x = 3\), then solving \(3^5\) by multiplying three by itself five times, we get \(243\).
Similarly, \(5^3\) means multiplying five by itself three times, equaling \(125\).
When dealing with exponentiation, it is essential to perform the multiplication accurately to get the correct result.
polynomial evaluation
Polynomials are algebraic expressions that consist of variables and coefficients combined using addition, subtraction, and multiplication.
Evaluating a polynomial involves substituting a specific value for the variable and then performing the arithmetic operations.
For instance, given a polynomial like \(3x^2 - 4x - 8\) and \(x = 3\), we substitute 3 into the polynomial.
This starts with replacing \(x\) with 3, resulting in \(3(3)^2 - 4(3) - 8\).
Breaking it down further, we calculate \(3(3)^2 = 27\), then \(4(3) = 12\), and finally combine to get \(27 - 12 - 8\), which simplifies to 7.
Polynomial evaluation helps in understanding the behavior of polynomials for different variables and is widely used in various fields of mathematics and science.
Evaluating a polynomial involves substituting a specific value for the variable and then performing the arithmetic operations.
For instance, given a polynomial like \(3x^2 - 4x - 8\) and \(x = 3\), we substitute 3 into the polynomial.
This starts with replacing \(x\) with 3, resulting in \(3(3)^2 - 4(3) - 8\).
Breaking it down further, we calculate \(3(3)^2 = 27\), then \(4(3) = 12\), and finally combine to get \(27 - 12 - 8\), which simplifies to 7.
Polynomial evaluation helps in understanding the behavior of polynomials for different variables and is widely used in various fields of mathematics and science.
Other exercises in this chapter
Problem 30
In the following exercises, simplify each expression. $$ 10+3[6-2(4-2)]-2^{4} $$
View solution Problem 31
In the following exercises, evaluate the following expressions. When \(x=2\), (a) \(x^{6}\) (b) \(4^{x}\) (c) \(2 x^{2}+3 x-7\)
View solution Problem 33
In the following exercises, evaluate the following expressions. When \(x=4, y=1\) \(x^{2}+3 x y-7 y^{2}\)
View solution Problem 34
In the following exercises, evaluate the following expressions. When \(x=3, y=2\) \(6 x^{2}+3 x y-9 y^{2}\)
View solution