Find the value of the expression: [tex]$\left(-\frac{2}{3}\right)^7 \div\left(-\frac{2}{3}\right)^4$[/tex]
Answer (2)
Apply the quotient of powers property: a m / a n = a m − n .
Rewrite the expression: ( − 3 2 ) 7 ÷ ( − 3 2 ) 4 = ( − 3 2 ) 7 − 4 .
Simplify the exponent: ( − 3 2 ) 3 .
Evaluate the expression: ( − 3 2 ) 3 = − 27 8 .
− 27 8
Explanation
Understanding the Problem We are asked to find the value of the expression ( − 3 2 ) 7 ÷ ( − 3 2 ) 4 . This involves dividing two exponential terms that have the same base.
Applying the Quotient of Powers Property To solve this, we will use the quotient of powers property, which states that when dividing exponential terms with the same base, we subtract the exponents: a m ÷ a n = a m − n .
Rewriting the Expression Applying this property to the given expression, we have: ( − 3 2 ) 7 ÷ ( − 3 2 ) 4 = ( − 3 2 ) 7 − 4 .
Simplifying the Exponent Now, we simplify the exponent: 7 − 4 = 3 .So, the expression becomes: ( − 3 2 ) 3 .
Evaluating the Expression Next, we evaluate the resulting expression. This means we need to calculate ( − 3 2 ) 3 = ( − 3 2 ) ⋅ ( − 3 2 ) ⋅ ( − 3 2 ) .
Calculating the Final Value Calculating the final value, we get: ( − 3 2 ) 3 = − 3 3 2 3 = − 27 8 .
Stating the Final Answer Therefore, the value of the expression ( − 3 2 ) 7 ÷ ( − 3 2 ) 4 is − 27 8 .
Examples
Imagine you're adjusting a recipe that calls for a certain amount of a spice blend. If you know the ratio of the blend's components and need to scale it down, you're essentially performing division with exponents. For instance, if the original recipe uses ( − 3 2 ) 7 units of the blend and you want to reduce it to ( − 3 2 ) 4 of the original amount, you would divide the two quantities, similar to the problem we solved. This ensures the flavors remain balanced even when the quantity changes. Understanding exponential division helps maintain proportions accurately in cooking, chemistry, and other fields where ratios are crucial.
To find the value of the expression ( − 3 2 ) 7 ÷ ( − 3 2 ) 4 , we apply the quotient of powers property to simplify it to ( − 3 2 ) 3 , which evaluates to − 27 8 .
;
