Find the value of the expression:
[tex]$27 a^0 c^3 \text { for } a=\frac{2}{3}, c=-\frac{1}{3}$[/tex]
Answer (2)
Substitute the given values into the expression: $27 a^0 c^3 = 27 (
\frac{2}{3})^0 (-
\frac{1}{3})^3$.
Simplify using a 0 = 1 : $27 (1) (-
\frac{1}{3})^3$.
Calculate $(-
\frac{1}{3})^3 = -
\frac{1}{27}$.
Multiply to find the final value: $27
\times (-
\frac{1}{27}) = -1 . T h e f ina l an s w er i s \boxed{{-1}}$.
Explanation
Understanding the Problem We are given the expression 27 a 0 c 3 and the values a = 3 2 and c = − 3 1 . Our goal is to find the value of the expression by substituting the given values.
Substituting the Values First, we substitute the given values of a and c into the expression: 27 a 0 c 3 = 27 ( 3 2 ) 0 ( − 3 1 ) 3
Simplifying the Expression Any non-zero number raised to the power of 0 is 1. Therefore, ( 3 2 ) 0 = 1 . So the expression becomes: 27 × 1 × ( − 3 1 ) 3
Calculating the Cube Now, we calculate ( − 3 1 ) 3 : ( − 3 1 ) 3 = − 3 1 × − 3 1 × − 3 1 = − 27 1
Substituting the Result Substitute this value back into the expression: 27 × 1 × ( − 27 1 ) = 27 × ( − 27 1 )
Final Calculation Finally, we multiply: 27 × ( − 27 1 ) = − 1
Examples
Understanding algebraic expressions and variable substitution is crucial in many real-world applications. For instance, in physics, you might use a formula to calculate the force acting on an object, where variables represent mass, acceleration, and other factors. By substituting specific values for these variables, you can determine the force. Similarly, in finance, you can use formulas to calculate interest earned on an investment, where variables represent the principal amount, interest rate, and time period. Substituting different values allows you to see how your investment grows under various conditions. This ability to manipulate and evaluate expressions is fundamental to problem-solving in various fields.
The value of the expression 27 a 0 c 3 when a = 3 2 and c = − 3 1 is − 1 . This is calculated by substituting values and simplifying using properties of exponents and arithmetic. The final result obtained is − 1 .
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