Rules When Using Variables As Exponents

Adding to the Exponent

• Adding to the exponent has the effect of multiplying the value of the expression by the base. For example, if the base is 4 and the exponent 2, the value of the expression is 16 because 4 X 4 = 16. If the exponent is a variable. it may change value. If the exponent is increased by 1, the new exponent is 3 and the new value is 64 because 4 X 4 X 4 = 64. The effect of increasing the exponent by 1 is the same as the effect of multiplying the value of the expression by 4 (the base) because 16 X 4 = 64.

Subtracting From the Exponent

• Subtracting from the exponent has the effect of dividing the value of the expression by the base. For example, if the base is 3 and the exponent is also 3, the value of the expression is 27 because 3 X 3 X 3 = 27. If the exponent is a variable. it may change value. If the exponent is decreased by 1, the new exponent is 2 and the new value is 9 because 3 X 3 = 9. The effect of decreasing the exponent by 1 is the same as the effect of dividing the value of the expression by 3 (the base) because 27 / 3 = 9.

Multiplying the Exponent

• Multiplying the exponent by a number has the effect of raising the value of the expression to a power. For example, if the base of the expression is 7 and the exponent is 2, the value of the expression is 49 because 7 X 7 = 49. If the exponent is a variable, it may change value. If the exponent is multiplied by 2, the new exponent is 4 and the new value is 2,401 because 7 X 7 X 7 X 7 = 2,401. The effect of multiplying the exponent by 2 is the same as squaring the expression because 49 X 49 = 2,401.

Dividing the Exponent

• Dividing the exponent by a number has the effect of taking the root of the expression. For example, if the base of the expression is 3 and the exponent is 4, the value of the expression is 81 because 3 X 3 X 3 X 3 = 81. If the exponent is a variable. it may change value. If the exponent is divided by 2, the new exponent is 2 and the new value is 9 because 3 X 3 = 9. The effect of dividing the exponent by 2 is the same as taking the square root of the expression because the square root of 81 is 9.

Summary of the Rules

• Adding to the exponent is the same as multiplying the expression by the base.
  Subtracting from the exponent is the same as dividing the expression by the base.
  Multiplying the exponent is the same as raising the expression to a power.
  Dividing the exponent is the same as taking a root of the expression.