Math Problem Statement

Simplify: (√25 / ∛64) + (256/625)^(-1/4) + 1 / (64/25)^(2/3)

Solution

Let's simplify the given expression step by step. The expression is:

25643+(256625)1/4+1(6425)2/3\frac{\sqrt{25}}{\sqrt[3]{64}} + \left(\frac{256}{625}\right)^{-1/4} + \frac{1}{\left(\frac{64}{25}\right)^{2/3}}

Step 1: Simplifying the first term

25643=54\frac{\sqrt{25}}{\sqrt[3]{64}} = \frac{5}{4}

  • 25=5\sqrt{25} = 5 since 5 is the square root of 25.
  • 643=4\sqrt[3]{64} = 4 since 43=644^3 = 64.

Step 2: Simplifying the second term

(256625)1/4\left(\frac{256}{625}\right)^{-1/4}

  • First, calculate the fourth root: 2564=4\sqrt[4]{256} = 4 (since 44=2564^4 = 256) and 6254=5\sqrt[4]{625} = 5 (since 54=6255^4 = 625).
  • The expression becomes (45)1\left(\frac{4}{5}\right)^{-1}, which is the reciprocal: 54\frac{5}{4}.

Step 3: Simplifying the third term

1(6425)2/3\frac{1}{\left(\frac{64}{25}\right)^{2/3}}

  • First, take the cube root: 643=4\sqrt[3]{64} = 4 and 253=5\sqrt[3]{25} = 5.
  • The expression becomes 45\frac{4}{5}.
  • Now square this: (45)2=1625\left(\frac{4}{5}\right)^2 = \frac{16}{25}.
  • The reciprocal is: 11625=2516\frac{1}{\frac{16}{25}} = \frac{25}{16}.

Step 4: Adding all terms

Now sum the results: 54+54+2516\frac{5}{4} + \frac{5}{4} + \frac{25}{16}

  • Convert all fractions to have a common denominator of 16: 54=2016\frac{5}{4} = \frac{20}{16}
  • Adding the fractions: 2016+2016+2516=6516\frac{20}{16} + \frac{20}{16} + \frac{25}{16} = \frac{65}{16}

Thus, the simplified expression is:

6516\boxed{\frac{65}{16}}

Would you like more details or explanations on any step? Here are some related questions to explore further:

  1. How do you find the reciprocal of a fraction raised to a negative exponent?
  2. What is the cube root of a number, and how do you calculate it?
  3. How can you add fractions with different denominators efficiently?
  4. How do you evaluate fractional exponents like x2/3x^{2/3}?
  5. Why does raising a fraction to a negative exponent involve taking the reciprocal?

Tip: When simplifying roots and exponents, it's often helpful to rewrite the terms in fractional form for easier manipulation.

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Math Problem Analysis

Mathematical Concepts

Algebra
Radicals
Exponents
Fractions

Formulas

√a (Square Root)
∛a (Cube Root)
(a/b)^n
Reciprocal of a fraction raised to a negative power

Theorems

Laws of exponents
Radical simplification

Suitable Grade Level

Grades 9-10