SENSEI AI
About App

The problem is to simplify expressions with negative exponents. There are 12 expressions in total that need to be simplified. I will solve each of these problems.

AlgebraExponentsSimplificationNegative ExponentsAlgebraic Expressions
2025/5/5

1. Problem Description

The problem is to simplify expressions with negative exponents. There are 12 expressions in total that need to be simplified. I will solve each of these problems.

2. Solution Steps

Problem 1: 3−23^{-2}
3−2=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}
Problem 2: 3−4⋅353^{-4} \cdot 3^{5}
Using the rule am⋅an=am+na^m \cdot a^n = a^{m+n}:
3−4⋅35=3−4+5=31=33^{-4} \cdot 3^{5} = 3^{-4+5} = 3^{1} = 3
Problem 3: 313−2\frac{3^{1}}{3^{-2}}
Using the rule aman=am−n\frac{a^m}{a^n} = a^{m-n}:
313−2=31−(−2)=31+2=33=27\frac{3^{1}}{3^{-2}} = 3^{1 - (-2)} = 3^{1+2} = 3^3 = 27
Problem 4: a−4a^{-4}
a−4=1a4a^{-4} = \frac{1}{a^4}
Problem 5: 4b−44b\frac{4b^{-4}}{\frac{4}{b}}
4b−44b=4⋅1b44b=4b44b=4b4⋅b4=4b4b4=1b3\frac{4b^{-4}}{\frac{4}{b}} = \frac{4 \cdot \frac{1}{b^4}}{\frac{4}{b}} = \frac{\frac{4}{b^4}}{\frac{4}{b}} = \frac{4}{b^4} \cdot \frac{b}{4} = \frac{4b}{4b^4} = \frac{1}{b^3}
Problem 6: 1c4c−2\frac{1c}{4c^{-2}}
c4c−2=14⋅cc−2=14c1−(−2)=14c1+2=14c3=c34\frac{c}{4c^{-2}} = \frac{1}{4} \cdot \frac{c}{c^{-2}} = \frac{1}{4} c^{1 - (-2)} = \frac{1}{4} c^{1+2} = \frac{1}{4}c^3 = \frac{c^3}{4}
Problem 7: (5d2)⋅(2d−5)(5d^{2}) \cdot (2d^{-5})
(5d2)⋅(2d−5)=5⋅2⋅d2⋅d−5=10⋅d2+(−5)=10d−3=10d3(5d^{2}) \cdot (2d^{-5}) = 5 \cdot 2 \cdot d^{2} \cdot d^{-5} = 10 \cdot d^{2+(-5)} = 10 d^{-3} = \frac{10}{d^3}
Problem 8: 9f43f−2\frac{9f^4}{3f^{-2}}
9f43f−2=93⋅f4f−2=3⋅f4−(−2)=3f4+2=3f6\frac{9f^4}{3f^{-2}} = \frac{9}{3} \cdot \frac{f^4}{f^{-2}} = 3 \cdot f^{4-(-2)} = 3f^{4+2} = 3f^6
Problem 9: (g−2)−2(g^{-2})^{-2}
Using the rule (am)n=am⋅n(a^m)^n = a^{m \cdot n}:
(g−2)−2=g(−2)⋅(−2)=g4(g^{-2})^{-2} = g^{(-2) \cdot (-2)} = g^{4}
Problem 10: (2h−3)2(2h^{-3})^{2}
(2h−3)2=22⋅(h−3)2=4⋅h−3⋅2=4h−6=4h6(2h^{-3})^{2} = 2^2 \cdot (h^{-3})^2 = 4 \cdot h^{-3 \cdot 2} = 4 h^{-6} = \frac{4}{h^6}
Problem 11: 2jk−52jk^{-5}
2jk−5=2j⋅1k5=2jk52jk^{-5} = 2j \cdot \frac{1}{k^5} = \frac{2j}{k^5}
Problem 12: (15m5)−3(\frac{15m}{5})^{-3}
(15m5)−3=(3m)−3=3−3⋅m−3=133⋅1m3=127⋅1m3=127m3(\frac{15m}{5})^{-3} = (3m)^{-3} = 3^{-3} \cdot m^{-3} = \frac{1}{3^3} \cdot \frac{1}{m^3} = \frac{1}{27} \cdot \frac{1}{m^3} = \frac{1}{27m^3}

3. Final Answer

1. $3^{-2} = \frac{1}{9}$

2. $3^{-4} \cdot 3^{5} = 3$

3. $\frac{3^{1}}{3^{-2}} = 27$

4. $a^{-4} = \frac{1}{a^4}$

5. $\frac{4b^{-4}}{\frac{4}{b}} = \frac{1}{b^3}$

6. $\frac{1c}{4c^{-2}} = \frac{c^3}{4}$

7. $(5d^{2}) \cdot (2d^{-5}) = \frac{10}{d^3}$

8. $\frac{9f^4}{3f^{-2}} = 3f^6$

9. $(g^{-2})^{-2} = g^{4}$

1

0. $(2h^{-3})^{2} = \frac{4}{h^6}$

1

1. $2jk^{-5} = \frac{2j}{k^5}$

1

2. $(\frac{15m}{5})^{-3} = \frac{1}{27m^3}$

Related problems in "Algebra"

We are asked to solve the quadratic equation $x^2 + x - 1 = 0$ for $x$.

Quadratic EquationsQuadratic FormulaRoots of Equations
2025/6/20

Solve the equation $\frac{x+1}{201} + \frac{x+2}{200} + \frac{x+3}{199} = -3$.

Linear EquationsEquation Solving
2025/6/20

The problem is to expand the given binomial expressions. The expressions are: 1. $(x + 1)(x + 3)$

Polynomial ExpansionBinomial ExpansionFOILDifference of Squares
2025/6/19

The problem is to remove the brackets and simplify the given expressions. I will solve question numb...

Algebraic ManipulationExpansionDifference of Squares
2025/6/19

We need to remove the brackets and collect like terms for the given expressions. I will solve proble...

Algebraic simplificationLinear expressionsCombining like termsDistribution
2025/6/19

The problem asks us to solve the equation $\lfloor 2x^3 - x^2 \rceil = 18x - 9$ for $x \in \mathbb{R...

EquationsCeiling FunctionReal NumbersCubic Equations
2025/6/19

The problem consists of 8 sub-problems. Each sub-problem contains an equation and a variable in pare...

Equation SolvingVariable IsolationFormula Manipulation
2025/6/19

The problem provides the equation of a parabola, $y = 3 - 2x - x^2$. We need to find the coordinates...

Quadratic EquationsParabolax-interceptTurning PointCoordinate Geometry
2025/6/19

The problem is to factorize the quadratic expression $2x^2 + 5x - 3$ completely.

Quadratic EquationsFactorizationPolynomials
2025/6/19

The problem consists of four parts. Part 1: Given the function $y = (2+x)(x-4)$, we need to sketch t...

Quadratic EquationsParabolaFactorizationGraphing
2025/6/19