Simplify the expression. [tex]$\left(b^3 g^4 i^{-5} j^0 k^4\right)^3$[/tex]
Answer (1)
Apply the power of a product rule: ( ab ) n = a n b n .
Apply the power of a power rule: ( a m ) n = a mn .
Simplify j 0 = 1 .
Rewrite the expression with the simplified terms and positive exponents: i 15 b 9 g 12 k 12 .
Explanation
Understanding the Expression We are asked to simplify the expression ( b 3 g 4 i − 5 j 0 k 4 ) 3 . This involves using the power of a product rule and the power of a power rule. We also need to remember that any non-zero term raised to the power of 0 is 1.
Applying Power of a Product Rule First, we apply the power of a product rule, which states that ( ab ) n = a n b n . This means we raise each term inside the parentheses to the power of 3: ( b 3 ) 3 ( g 4 ) 3 ( i − 5 ) 3 ( j 0 ) 3 ( k 4 ) 3
Applying Power of a Power Rule Next, we apply the power of a power rule, which states that ( a m ) n = a mn . Applying this rule to each term, we get: b 3 × 3 g 4 × 3 i − 5 × 3 j 0 × 3 k 4 × 3 This simplifies to: b 9 g 12 i − 15 j 0 k 12
Simplifying j^0 Now, we simplify j 0 . Any non-zero term raised to the power of 0 is 1, so j 0 = 1 . Our expression becomes: b 9 g 12 i − 15 ( 1 ) k 12 Which is the same as: b 9 g 12 i − 15 k 12
Rewriting with Positive Exponents Finally, we rewrite the expression with positive exponents. To do this, we move i − 15 to the denominator, which changes the sign of the exponent: i 15 b 9 g 12 k 12
Final Answer Therefore, the simplified expression is i 15 b 9 g 12 k 12 .
Examples
Understanding how to simplify expressions with exponents is crucial in many areas of science and engineering. For example, in physics, when dealing with quantities like volume or area that scale with length, simplifying expressions with exponents helps in understanding how these quantities change with changes in length. If you have a cube with side length b 3 , then the volume of the cube is ( b 3 ) 3 = b 9 . Similarly, in computer science, when analyzing the complexity of algorithms, exponential expressions often arise, and simplifying them helps in understanding the efficiency of the algorithm.
