Final answer:
To write the square root of 56 in rational exponent form, it is converted to 56^1/2. For Part B, X3 is already in exponent form. Part C is simplified by converting radicals to rational exponents; thus, √8(√8 + 8^2) becomes 8(8 + 64) or 576.
Step-by-step explanation:
Writing an equivalent expression in rational exponent form involves converting radical expressions to expressions with exponents. This allows us to work with them more easily in algebraic operations.
Part A: √56
To convert the square root of 56 into a rational exponent form, recall that a square root is equivalent to raising a number to the power of ½. Thus:
√56 = 561/2
Part B: X3
The cube of a variable is simply written using an exponent:
X3 (no conversion is necessary for simple exponentiation).
Part C: √8(√8 + 82)
First, convert each radical to its equivalent rational exponent:
√8 = 81/2 and √8 = 81/2. Now, we can rewrite and simplify the expression:
(81/2)(81/2 + 82)
Next, when multiplying exponential terms with the same base, we add the exponents:
81/2 × 81/2 = 8(1/2 + 1/2) = 81
Finally, we have:
8(8 + 82)
To further simplify, we can perform the operations inside the parenthesis:
8(8 + 64)
8(72)
The fully simplified expression in rational exponent form then is:
81 × 72 = 576