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1. \(4×2=2^2 2^1=2^{2+1}=2^3\)
2. \(3×27=3^1 3^3=3^{1+3}=3^4\)
3. \(2^{3n} \times 2^{5n}=2^{3n+5n}=2^{8n}\)
4. \(3^{4n} \div 3^{5n}=3^{4n-5n}=3^{-n}={1 \over 3^n}\)
5. \(a^{1 \over 2}(9a)^{1 \over 2}=a^{1\over 2} \times 9^{1 \over 2}a^{1 \over 2}\)
\(=a^{1 \over 2}(3a^{1 \over 2})\)
\(=3a^{{1 \over 2} + {1 \over 2}}\)
\(=3a\)
Take note parentheses plays an important role. For example, \(-2^2\) =-4 but \(-2^2\)=4. Also, \(ab^2\)=a×b×b but \(ab^2\)=a×a×b×b.
Example
Simplify the following.
\((a^4b^3c^2) \times (ab^5c^2) \)
\(=a^{4+1}b^{3+5}c^{2+2}\)
\(=a^5b^8c^4\)
\(x^3y^2z^3 \div x^2y^5\)
\(=x^{3-2}y^{2-5}z^{3-0}\)
\(=x^1y^{-3}z^3 \)
\(={xz^3 \over y^3}\)
1. Solving equations
You can use the following steps to solve equations involving indices:
- express both sides of the equation in the same base
- equate the indices and solve
Example
1. Solve for x the equation \(4^x=64.\)
\(4^x=4^3\)
\(x=3\)
2. Solve for x the equation \((2^{3x})(2^{x-1})=32\)
\(2^{3x+x-1}=2^5\)
\(2^{4x-1}=2^5\)
\(4x-1=5\)
\(4x=6\)
\(x={3 \over 2}\)
3. Solve for x the equation \({(16^x)(16^x) \over 16^x}=8\).
\({16^{2x} \over 16^x}=8\)
\(16^{2x-x}=8\)
\(16^x=8\)
\((2^{4x})=2^3\)
\(4x=3\)
\(x={3 \over 4}\)
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