Let's simplify the expressions provided step by step.
Problem 2.1.1:
Expression: \( \frac{2x^{-9}}{3x^{-3}} \)
Step 1: Rewrite each term in the expression to eliminate negative exponents: \[ x^{-9} = \frac{1}{x^9} \quad \text{and} \quad x^{-3} = \frac{1}{x^3} \] So we can rewrite the expression: \[ \frac{2x^{-9}}{3x^{-3}} = \frac{2}{3} \cdot \frac{1}{x^9} \cdot x^3 = \frac{2}{3} \cdot \frac{x^3}{x^9} \]
Step 2: Apply the laws of exponents: \[ \frac{x^3}{x^9} = x^{3-9} = x^{-6} \] Thus, substituting back, we have: \[ \frac{2}{3} \cdot x^{-6} = \frac{2}{3x^6} \] Final answer for 2.1.1: \[ \frac{2}{3x^6} \]
Problem 2.1.2:
Expression: \( \frac{12 + 4 \cdot 75}{(5+3)(5-3)} \)
Step 1: Calculate the denominator: \[ (5 + 3)(5 - 3) = 8 \cdot 2 = 16 \]
Step 2: Calculate the numerator: First, simplify \( 4 \cdot 75 \): \[ 4 \cdot 75 = 300 \] Now, add to 12: \[ 12 + 300 = 312 \]
Step 3: Now put everything together: \[ \frac{12 + 4 \cdot 75}{(5+3)(5-3)} = \frac{312}{16} \]
Step 4: Simplify \( \frac{312}{16} \): Divide both the numerator and the denominator by 16: \[ 312 \div 16 = 19.5 \quad \text{(but we will keep it in a proper fraction form)} \]
Step 5: Converting to a proper simplified fraction: Both numbers can still be divided by 4: \[ \frac{312 \div 4}{16 \div 4} = \frac{78}{4} \] Again, dividing by 2: \[ \frac{78 \div 2}{4 \div 2} = \frac{39}{2} \]
Final answer for 2.1.2: \[ \frac{39}{2} \]
Summary of Answers:
- 2.1.1: \( \frac{2}{3x^6} \)
- 2.1.2: \( \frac{39}{2} \)
1 answer