“Rewrite ( 1.5 ) as ( \frac{3}{2} ).” Ms. Vega leans in. “The rule holds for all rational exponents. Now: The base is ( \frac{1}{4} ). Negative exponent → flip it: ( 4^{3/2} ). Denominator 2 → square root of 4 is 2. Numerator 3 → cube 2 to get 8. Done.”
“( 27^{-2/3} ) whispers: ‘I was once ( 27^{2/3} ), but someone took my reciprocal.’ So first, undo the mirror: ( 27^{-2/3} = \frac{1}{27^{2/3}} ). Then apply the fraction rule: cube root of 27 is 3, square is 9. So answer: ( \frac{1}{9} ).” Fractional Exponents Revisited Common Core Algebra Ii
Ms. Vega pushes her mug aside. “You’re thinking like a robot. Let’s tell a story.” “Rewrite ( 1
“That’s not a fraction — it’s a decimal,” Eli protests. Now: The base is ( \frac{1}{4} )
“Last boss,” Ms. Vega taps the page: ( \left(\frac{1}{4}\right)^{-1.5} ).
Eli writes: ( \left(\frac{1}{4}\right)^{-1.5} = 8 ). He stares. “That’s beautiful.”