Given k = - 3, find the value of k3 + 3k: A. - 36 B. 36 C. 37 D. - 37
A
Understanding the Problem: Evaluating Algebraic Expressions
The question asks us to find the value of an algebraic expression, \(k^3 + 3k\), given a specific value for the variable \(k\). To solve this, we need to substitute the given value of \(k\) into the expression and perform the necessary arithmetic operations.
The given value is \(k = -3\). The expression we need to evaluate is \(k^3 + 3k\).
Step-by-Step Solution: Substituting and Calculating
We will substitute \(k = -3\) into the expression \(k^3 + 3k\).
- First, calculate the value of \(k^3\). Substitute \(k = -3\):
- \(k^3 = (-3)^3\)
- To calculate \( (-3)^3 \), we multiply -3 by itself three times:
- \( (-3)^3 = (-3) \times (-3) \times (-3) \)
- \( (-3) \times (-3) = 9 \) (The product of two negative numbers is positive)
- \( 9 \times (-3) = -27 \) (The product of a positive and a negative number is negative)
- So, \(k^3 = -27\).
- Next, calculate the value of \(3k\). Substitute \(k = -3\):
- \(3k = 3 \times (-3)\)
- \(3 \times (-3) = -9\) (The product of a positive and a negative number is negative)
- So, \(3k = -9\).
- Now, add the values of \(k^3\) and \(3k\):
- \(k^3 + 3k = (-27) + (-9)\)
- Adding a negative number is the same as subtracting the corresponding positive number:
- \( (-27) + (-9) = -27 - 9 \)
- \( -27 - 9 = -36 \)
Therefore, the value of \(k^3 + 3k\) when \(k = -3\) is \(-36\).
Checking the Options
Let's compare our result with the given options:
- A. -36
- B. 36
- C. 37
- D. -37
Our calculated value, -36, matches option A.
Revision Table: Key Concepts
| Concept | Description | Example |
|---|---|---|
| Algebraic Expression | A mathematical phrase that contains numbers, variables, and operators. | \(k^3 + 3k\) |
| Variable | A symbol, usually a letter, representing a quantity that may change. | \(k\) in \(k^3 + 3k\) |
| Evaluating an Expression | Finding the value of an expression by substituting the given values for variables and performing the operations. | Substituting \(k = -3\) into \(k^3 + 3k\) |
| Exponentiation | Raising a base number to a power (exponent). Indicates repeated multiplication. | \(k^3\) means \(k \times k \times k\) |
| Rules of Signs | Rules for multiplying and adding positive and negative numbers. | \( (-) \times (-) = (+) \), \( (-) + (-) = (-) \) |
Additional Information: Working with Negative Numbers and Exponents
When evaluating expressions involving negative numbers and exponents, it's important to pay close attention to the rules of signs.
- When a negative number is raised to an odd power (like 3), the result is negative. For example, \( (-3)^3 = -27 \).
- When a negative number is raised to an even power (like 2 or 4), the result is positive. For example, \( (-3)^2 = (-3) \times (-3) = 9 \).
- When multiplying a positive number by a negative number, the result is negative. For example, \( 3 \times (-3) = -9 \).
- When adding two negative numbers, the result is a negative number with a magnitude equal to the sum of their magnitudes. For example, \( -27 + (-9) = -36 \).
Practicing these rules helps avoid common errors when evaluating algebraic expressions.
Important Questions from Polynomials
If (x + y) 3+ 8 (x - y) 3= (3x + Ay) (3x 2+ Bxy + Cy 2), then the value of A + B + C is:
Given that x 8- 34x 4+ 1 = 0, x > 0. What is the value of (x 3+ x -3 )?
If \(x - \frac 3 x = 6,\; x \ne 0,\) then the value of \(\frac {x^4 - \frac {27}{x^2}}{x^2 - 3x - 3}\) is:
If \(x\left(3 - \frac 2 x\right) = \frac 3 x,\) then the value of \(x^3 - \frac 1 {x^3}\) is equal to:
The coefficient of x in (x – 3y) 3is:
