What is the value of 64x3 + 38x2y + 20xy2 + y3, when x = 3 and y = - 4?
1256
Evaluating Algebraic Expressions with Given Values
The question asks us to find the value of the algebraic expression \(64x^3 + 38x^2y + 20xy^2 + y^3\) when specific values for \(x\) and \(y\) are provided. We are given \(x = 3\) and \(y = -4\). To solve this, we need to substitute these values into the expression and perform the necessary calculations.
Steps to Evaluate the Expression
We will follow these steps:
- Write down the given algebraic expression.
- Substitute the given values of \(x\) and \(y\) into the expression.
- Calculate the value of each term in the expression.
- Add or subtract the values of the terms to find the final value of the expression.
The given expression is: \(64x^3 + 38x^2y + 20xy^2 + y^3\)
The given values are: \(x = 3\) and \(y = -4\)
Substitution and Calculation
Substitute \(x = 3\) and \(y = -4\) into the expression:
\(64(3)^3 + 38(3)^2(-4) + 20(3)(-4)^2 + (-4)^3\)
Now, let's calculate each term separately:
- Term 1: \(64x^3 = 64(3)^3\)
- First, calculate \(3^3\): \(3 \times 3 \times 3 = 9 \times 3 = 27\)
- Now, multiply by 64: \(64 \times 27\)
- \(64 \times 27 = 1728\)
- Term 2: \(38x^2y = 38(3)^2(-4)\)
- First, calculate \(3^2\): \(3 \times 3 = 9\)
- Now, multiply 38 by 9: \(38 \times 9 = 342\)
- Finally, multiply by -4: \(342 \times (-4) = -1368\)
- Term 3: \(20xy^2 = 20(3)(-4)^2\)
- First, calculate \((-4)^2\): \((-4) \times (-4) = 16\)
- Now, multiply 20 by 3: \(20 \times 3 = 60\)
- Finally, multiply by 16: \(60 \times 16 = 960\)
- Term 4: \(y^3 = (-4)^3\)
- Calculate \((-4)^3\): \((-4) \times (-4) \times (-4) = 16 \times (-4) = -64\)
Now, we add the values of the calculated terms:
\(1728 + (-1368) + 960 + (-64)\)
\(1728 - 1368 + 960 - 64\)
Perform the additions and subtractions:
\(1728 - 1368 = 360\)
\(960 - 64 = 896\)
Wait, let's combine the terms as they appear:
\(1728 - 1368 = 360\)
\(360 + 960 = 1320\)
\(1320 - 64 = 1256\)
So, the value of the expression \(64x^3 + 38x^2y + 20xy^2 + y^3\) when \(x = 3\) and \(y = -4\) is 1256.
Let's summarize the term values in a table:
| Term | Calculation | Value |
|---|---|---|
| \(64x^3\) | \(64(3)^3 = 64 \times 27\) | 1728 |
| \(38x^2y\) | \(38(3)^2(-4) = 38 \times 9 \times (-4) = 342 \times (-4)\) | -1368 |
| \(20xy^2\) | \(20(3)(-4)^2 = 20 \times 3 \times 16 = 60 \times 16\) | 960 |
| \(y^3\) | \((-4)^3\) | -64 |
Total value = \(1728 + (-1368) + 960 + (-64) = 1728 - 1368 + 960 - 64 = 1256\)
Conclusion
By substituting \(x = 3\) and \(y = -4\) into the expression \(64x^3 + 38x^2y + 20xy^2 + y^3\) and performing the calculations, we found the value of the expression to be 1256.
Revision Table: Key Concepts for Expression Evaluation
| Concept | Description | Example |
|---|---|---|
| Algebraic Expression | A mathematical phrase that can contain numbers, variables, and operators. | \(2x + 5y - 3\) |
| Variable | A symbol (usually a letter) representing a quantity that may change. | \(x\), \(y\) in \(x+y=7\) |
| Term | A single number or variable, or numbers and variables multiplied together. Terms are separated by + or - signs. | In \(3x^2 + 2xy - 5\), the terms are \(3x^2\), \(2xy\), and \(-5\). |
| Evaluating an Expression | Finding the numerical value of an expression by substituting given values for the variables and performing the operations. | Evaluating \(2x+3\) for \(x=4\): \(2(4)+3 = 8+3=11\). |
| Order of Operations | The rule that states the sequence in which multiple operations in an expression should be performed (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Often remembered by PEMDAS or BODMAS. | To calculate \(2 + 3 \times 4\), multiply first: \(2 + 12 = 14\). |
Additional Information: Polynomial Expressions
The expression \(64x^3 + 38x^2y + 20xy^2 + y^3\) is a type of polynomial, specifically a homogeneous polynomial because every term has the same total degree (the sum of the exponents of the variables in the term). In this case, the degree of each term is 3 (\(x^3\), \(x^2y\) - degree 2+1=3, \(xy^2\) - degree 1+2=3, \(y^3\) - degree 3).
Evaluating polynomials is a fundamental skill in algebra and is used in many areas of mathematics and science. When evaluating, it's crucial to pay close attention to the signs of the numbers, especially when dealing with negative values and exponents, as seen with \(y = -4\) in this problem.
- When a negative number is raised to an even power, the result is positive (e.g., \((-4)^2 = 16\)).
- When a negative number is raised to an odd power, the result is negative (e.g., \((-4)^3 = -64\)).
Accuracy in calculation is key to getting the correct value of the expression.
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