Math Problem Statement

Simplify the following expressions: (2x)^3, (4y)^2, (4x^2)^3, (6y^5)^2, (-2x)^4, (-3y)^3, (2x^3/y)^3, (2x^2/y)^6.

Solution

The image contains the following set of expressions to be simplified:

Task: "Suprastinkite reiškinį" translates to "Simplify the expression."

  1. (2x)3(2x)^3
  2. (4y)2(4y)^2
  3. (4x2)3(4x^2)^3
  4. (6y5)2(6y^5)^2
  5. (2x)4(-2x)^4
  6. (3y)3(-3y)^3
  7. (2x3y)3\left(\frac{2x^3}{y}\right)^3
  8. (2x2y)6\left(\frac{2x^2}{y}\right)^6

Let’s simplify each one step-by-step:

a) (2x)3(2x)^3

(2x)3=23x3=8x3(2x)^3 = 2^3 \cdot x^3 = 8x^3

b) (4y)2(4y)^2

(4y)2=42y2=16y2(4y)^2 = 4^2 \cdot y^2 = 16y^2

c) (4x2)3(4x^2)^3

(4x2)3=43(x2)3=64x6(4x^2)^3 = 4^3 \cdot (x^2)^3 = 64x^6

d) (6y5)2(6y^5)^2

(6y5)2=62(y5)2=36y10(6y^5)^2 = 6^2 \cdot (y^5)^2 = 36y^{10}

e) (2x)4(-2x)^4

(2x)4=(2)4x4=16x4(-2x)^4 = (-2)^4 \cdot x^4 = 16x^4

f) (3y)3(-3y)^3

(3y)3=(3)3y3=27y3(-3y)^3 = (-3)^3 \cdot y^3 = -27y^3

g) (2x3y)3\left( \frac{2x^3}{y} \right)^3

(2x3y)3=23(x3)3y3=8x9y3\left( \frac{2x^3}{y} \right)^3 = \frac{2^3 \cdot (x^3)^3}{y^3} = \frac{8x^9}{y^3}

h) (2x2y)6\left( \frac{2x^2}{y} \right)^6

(2x2y)6=26(x2)6y6=64x12y6\left( \frac{2x^2}{y} \right)^6 = \frac{2^6 \cdot (x^2)^6}{y^6} = \frac{64x^{12}}{y^6}

Final Simplified Results:

a) 8x38x^3
b) 16y216y^2
c) 64x664x^6
d) 36y1036y^{10}
e) 16x416x^4
f) 27y3-27y^3
g) 8x9y3\frac{8x^9}{y^3}
h) 64x12y6\frac{64x^{12}}{y^6}

Would you like more details or have any questions?


Related Questions:

  1. How do you apply the power of a product rule in exponentiation?
  2. What happens when you raise a negative number to an even power versus an odd power?
  3. How can you simplify fractions with exponents?
  4. How do you handle exponents in expressions with multiple variables?
  5. Can you explain how to deal with negative exponents?

Tip: Always apply the exponent to both the coefficient and the variable when simplifying expressions.

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Algebraic Expressions
Powers of Variables
Negative and Positive Exponents

Formulas

Power of a product: (ab)^n = a^n * b^n
Power of a quotient: (a/b)^n = a^n / b^n
Negative exponent: a^(-n) = 1/a^n

Theorems

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Suitable Grade Level

Grades 7-9