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Simplify expressions using exponent laws

Simplifying expressions with exponent laws involves interpreting powers as repeated multiplication and preserving equivalent values by combining like bases: products add exponents, quotients subtract exponents, and powers of powers multiply exponents, with exponents distributed across products or quotients when appropriate. The scope includes numerical and algebraic expressions with integer exponents, including zero and negative exponents for nonzero bases, while distinguishing multiplication of powers from addition; fractional exponents and more advanced generalizations are not included.

Detailed Explanation: Simplify expressions using exponent laws

Exponents show repeated multiplication. When simplifying, use exponent laws only when the expressions have the same base:

  • Multiply like bases: amâ‹…an=am+na^m\cdot a^n=a^{m+n}
  • Divide like bases: aman=am−n\dfrac{a^m}{a^n}=a^{m-n}
  • Power of a power: (am)n=amn(a^m)^n=a^{mn}
  • Power of a product: (ab)n=anbn(ab)^n=a^n b^n

Remember: exponents are added when powers are multiplied, not when terms are added.

Simplify:

(2x3)2â‹…x44x2\frac{(2x^3)^2\cdot x^4}{4x^2}

Step 1: Simplify the power of a product and a power of a power.

(2x3)2=22(x3)2=4x6(2x^3)^2=2^2(x^3)^2=4x^6

So the expression becomes

4x6â‹…x44x2\frac{4x^6\cdot x^4}{4x^2}

Step 2: Combine the powers in the numerator.

Since x6x^6 and x4x^4 are being multiplied, add their exponents:

x6â‹…x4=x6+4=x10x^6\cdot x^4=x^{6+4}=x^{10}

Now we have

4x104x2\frac{4x^{10}}{4x^2}

Step 3: Divide the coefficients and subtract the exponents.

44=1andx10x2=x10−2=x8\frac{4}{4}=1 \qquad\text{and}\qquad \frac{x^{10}}{x^2}=x^{10-2}=x^8

Therefore,

x8\boxed{x^8}

This simplification assumes x≠0x\ne 0, because the original expression has x2x^2 in the denominator.

Learn by doing: Simplify expressions using exponent laws

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    Exponents - Multiplication - Positive by Positive to Positive

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