Rewriting exponential expressions is a fundamental skill in algebra. We use exponent rules to simplify, expand, or change the form of expressions. Key techniques include changing the base, applying product and quotient rules, and handling negative exponents.
The first technique for rewriting exponential expressions is changing the base. We express the base as a power of a smaller or more common base. For example, eight can be written as two to the third power, so eight to the x becomes two to the three x. This technique reveals patterns and makes calculations much easier.
The second technique involves applying core exponent rules. The product rule adds exponents when multiplying same bases. The quotient rule subtracts exponents when dividing. The power rule multiplies exponents when raising a power to another power. These rules form the foundation of exponential manipulation.
Handling negative and rational exponents is crucial for rewriting expressions. Negative exponents become positive by taking reciprocals. For example, seven to the negative two equals one over seven squared. Rational exponents involve roots and powers, like sixteen to the three halves equals four cubed, which is sixty-four.
Let's solve a complex example using all our techniques. We start with twenty-seven to the x times nine to the negative x, divided by three to the x plus one. First, we change bases: twenty-seven becomes three cubed, nine becomes three squared. Then we apply the power rule and combine exponents. Finally, we use the quotient rule to get three to the negative one, which equals one-third.