### Perform the Indicated Operations (without a calculator)1. $-6 + (-5)^3$ 2. $(-2)^2 + 4.5((-1) - 3(-3 + 11))$**Note:** Recall the order of operations and operations with signed numbers.**Extra:** Create/solve another numerical expression with all arithmetic operations.This exercise focuses on practicing mathematical operations, including exponentiation, addition, and multiplication, with particular attention to signed numbers and the order of operations (PEMDAS/BODMAS). **Simplify and write the answer in exponential notation for each problem. Use and indicate an appropriate power rule(s) for each expression.**a) $ 5^2 \cdot 5^3 $b) $ \frac{3^4}{3^3} $c) $ (6^3)^2 $d) $ \frac{(3x^2 y^4)^2}{x^2} $**Note:** Recall all (five) power rules.**Extra:** Construct/solve an expression where all power rules are to be applied.---**Explanation:**- **Power Rules to Remember:** 1. $ a^m \cdot a^n = a^{m+n} $ 2. $ \frac{a^m}{a^n} = a^{m-n} $ 3. $ (a^m)^n = a^{m \cdot n} $ 4. $ a^{-n} = \frac{1}{a^n} $ 5. $ (ab)^n = a^n \cdot b^n $ - **Application on Problems:** *For each part, apply these rules to simplify the expressions.*

3. -6+-53 Here -53 =-5×-5×-5=-125 Now -6+-53=-6-125=-131 Add the two numbers and put the same sign…

Simplify and write the answer in exponential notation for each problem. Use and indicate an appropriate power rule(s) for each expression. 3* (3x²y*)' 52.5 4 a) (6') d) b) 33 Note: Recall all (five) power rules. Extra: Construct/solve an expression whereall power rules to be applied.

Simplify and write the answer in exponential notation for each problem. Use and indicate an appropriate power rule(s) for each expression. 3* (3x²y*)' 52.5 4 a) (6') d) b) 33 Note: Recall all (five) power rules. Extra: Construct/solve an expression whereall power rules to be applied.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

### Perform the Indicated Operations (without a calculator)

1. $-6 + (-5)^3$
2. $(-2)^2 + 4.5((-1) - 3(-3 + 11))$

**Note:** Recall the order of operations and operations with signed numbers.

**Extra:** Create/solve another numerical expression with all arithmetic operations.

This exercise focuses on practicing mathematical operations, including exponentiation, addition, and multiplication, with particular attention to signed numbers and the order of operations (PEMDAS/BODMAS).

**Simplify and write the answer in exponential notation for each problem. Use and indicate an appropriate power rule(s) for each expression.**

a) $ 5^2 \cdot 5^3 $

b) $ \frac{3^4}{3^3} $

c) $ (6^3)^2 $

d) $ \frac{(3x^2 y^4)^2}{x^2} $

**Note:** Recall all (five) power rules.

**Extra:** Construct/solve an expression where all power rules are to be applied.

---

**Explanation:**

- **Power Rules to Remember:**
1. $ a^m \cdot a^n = a^{m+n} $
2. $ \frac{a^m}{a^n} = a^{m-n} $
3. $ (a^m)^n = a^{m \cdot n} $
4. $ a^{-n} = \frac{1}{a^n} $
5. $ (ab)^n = a^n \cdot b^n $

- **Application on Problems:**

*For each part, apply these rules to simplify the expressions.*

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