Math Problem Statement

Nyatakan ke bentuk paling sederhana dalam pangkat positif: (x^2y^{-3}z^{-1}/x^3y^5z^{-4})^2; Tentukan himpunan penyelesaian dari persamaan eksponen berikut: 3^{3x^2+2x-1} = 3^{2x+1}; Hitunglah nilai dari: 125^{1/3} + 24^{3/2} - 128^{2/7}; Sederhanakan bentuk akar berikut: 3√150 + 5√54 - 8√96; Rasionalkan penyebut dari pecahan berikut: √32/√12

Solution

Here is a breakdown of the questions from the image you've uploaded:


1. Nyatakan ke bentuk paling sederhana dalam pangkat positif:

(x2y3z1x3y5z4)2\left( \frac{x^2y^{-3}z^{-1}}{x^3y^5z^{-4}} \right)^2

Simplify this expression using positive exponents.


2. Tentukan himpunan penyelesaian dari persamaan eksponen berikut:

33x2+2x1=32x+13^{3x^2+2x-1} = 3^{2x+1}

Find the solution set for the given exponential equation.


3. Hitunglah nilai dari:

12513+243212827125^{\frac{1}{3}} + 24^{\frac{3}{2}} - 128^{\frac{2}{7}}

Calculate the value of this expression.


4. Sederhanakan bentuk akar berikut:

a) 3150+5548963\sqrt{150} + 5\sqrt{54} - 8\sqrt{96}

b) 32582128\frac{\sqrt{32} - 5\sqrt{8}}{2\sqrt{128}}

Simplify these square roots.


5. Rasionalkan penyebut dari pecahan berikut:

a) 3212\frac{\sqrt{32}}{\sqrt{12}}

b) 33+2\frac{3}{3+\sqrt{2}}

Rationalize the denominator in these two fractions.


Let me know if you'd like to go through each question one by one or if you need specific help with any question!

Would you like the solutions to these questions in detail? Here are 5 questions that can deepen your understanding:

  1. What rules apply when simplifying expressions with exponents and radicals?
  2. How do you solve exponential equations using the property of equal bases?
  3. What is the process for converting radicals to simplified form?
  4. How do you approach rationalizing denominators with square roots?
  5. Can you identify when a cube root and square root need simplification before adding them together?

Tip: Always check if you can factor or simplify before applying exponent rules to make calculations easier!

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Math Problem Analysis

Mathematical Concepts

Exponentiation
Simplification of Radicals
Exponential Equations
Rationalizing Denominators

Formulas

a^m / a^n = a^(m-n)
(a^m)^n = a^(m*n)
a^(m/n) = n√(a^m)
Rationalization formula for square roots: Multiply by conjugate

Theorems

Properties of Exponents
Radical Simplification Theorem
Laws of Exponents
Rationalizing Denominators

Suitable Grade Level

Grade 9-10