5 Easy Steps to Solve Parentheses Raised to a Power

Additionally, the expression within the parentheses can be simplified before raising it to the power. For example, if the expression within the parentheses is a sum or difference, it can be simplified using the distributive property. If the expression within the parentheses is a product or quotient, it can be simplified using the associative and commutative properties.

However, there are some cases where it is not possible to simplify the expression within the parentheses. In these cases, it is necessary to use the binomial theorem to expand the expression. The binomial theorem is a formula that can be used to expand the expression (a + b)^n, where n is a positive integer. The formula is as follows:

“`
(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k
“`

Where \binom{n}{k} is the binomial coefficient, which is given by the formula:

“`
\binom{n}{k} = \frac{n!}{k!(n-k)!}
“`

Simplification of Expressions

Expressions containing parentheses raised to a power can be simplified using the following steps:

To simplify an expression with parentheses raised to a power, follow these steps:

Step 1: Identify the terms with parentheses raised to a power.

For example, in the expression (a + b)^2, the term (a + b) is enclosed in parentheses and raised to the power of 2.

Step 2: Distribute the power to each term within the parentheses.

In the above example, we distribute the power of 2 to each term within the parentheses (a + b), resulting in:

“`
(a + b)^2 = a^2 + 2ab + b^2
“`

Step 3: Simplify the resulting expression.

Combine like terms and simplify any resulting fractions or radicals. For example,

“`
(x – 2)(x + 5) = x^2 + 5x – 2x – 10 = x^2 + 3x – 10
“`

The steps outlined above can be applied to simplify any expression containing parentheses raised to a power.

Expression	Simplified Form
(x + y)^3	x^3 + 3x^2y + 3xy^2 + y^3
(2a – b)^4	16a^4 – 32a^3b + 24a^2b^2 – 8ab^3 + b^4
(x – 3y)^5	x^5 – 15x^4y + 90x^3y^2 – 270x^2y^3 + 405xy^4 – 243y^5

Distributing Exponents

When parentheses are raised to a power, we can distribute the exponent to each term within the parentheses. This means that the exponent applies not only to the entire expression within the parentheses but also to each individual term. For instance:

(x + y)^2 = x^2 + 2xy + y^2

In this expression, the exponent 2 is distributed to both x and y. Similarly, for more complex expressions:

(a + b + c)^3 = a^3 + 3a^2(b + c) + 3ab^2 + 6abc + b^3 + 3bc^2 + c^3

The following table provides a summary of the rules for distributing exponents:

Expression	Expanded Form
(ab)^n	aⁿbⁿ
(a + b)^n	aⁿ + n(a^n-1b) + n(a^n-2b²) + … + bⁿ
(a – b)^n	aⁿ – n(a^n-1b) + n(a^n-2b²) – … + (-1)ⁿbⁿ

Negative Exponents and Parentheses

When dealing with negative exponents and parentheses, it’s important to remember the following rule:

(a^-b) = 1/(a^b)

This means that when you have a negative exponent inside parentheses, you can rewrite it by moving the exponent to the denominator and changing the sign to positive.

For example:

(x^-2) = 1/(x^2)

(y^-3) = 1/(y^3)

Using this rule, you can simplify expressions with negative exponents and parentheses. For instance:

(x^-2)^3 = (1/(x^2))^3 = 1/(x^6)

((-y)^-4)^2 = (1/((-y)^4))^2 = 1/((y)^8) = 1/(y^8)

To fully understand this concept, let’s delve deeper into the mathematical operations involved:

Raising a Parenthesis to a Negative Exponent: When you raise a parenthesis to a negative exponent, you are essentially taking the reciprocal of the original expression. This means that (a^-b) is equal to 1/(a^b).

Simplifying Expressions with Negative Exponents: To simplify expressions with negative exponents, you can use the rule (a^-b) = 1/(a^b). This allows you to rewrite the expression with a positive exponent in the denominator.

Applying the Rule to Real-World Scenarios: Negative exponents and parentheses are commonly used in various fields, including physics and engineering. For example, in physics, the inverse square law is often expressed using negative exponents. In engineering, negative exponents are used to represent quantities that are reciprocals of other quantities.

Nested Exponents

When exponents are raised to another power, we have nested exponents. To simplify such expressions, we use the following rules:

Power of a Power Rule

To raise a power to another power, multiply the exponents:

“`
(a^m)^n = a^(m*n)
“`

Power of a Product Rule

To raise a product to a power, raise each factor to that power:

“`
(ab)^n = a^n * b^n
“`

Power of a Quotient Rule

To raise a quotient to a power, raise the numerator and denominator separately to that power:

“`
(a/b)^n = a^n / b^n
“`

Raising Powers to Fractional Exponents

When raising a power to a fractional exponent, it’s equivalent to extracting the root of that power:

“`
(a^m)^(1/n) = a^(m/n)
“`

Fractional Exponents and Parentheses

When a parenthetical expression is raised to a fractional exponent, it is important to apply the exponent to both the parenthetical expression and the individual terms within it. For example:

(a + b)^1/2 = √(a + b)

(a – b)^1/2 = √(a – b)

(ax² + bx)^1/2 = √(ax² + bx)

Applying Fractional Exponents to Individual Terms

In some cases, it may be necessary to apply fractional exponents to individual terms within a parenthetical expression. In such cases, it is important to remember that the exponent should be applied to the entire term, including any coefficients or variables.

For example:

(2ax² + bx)^1/2 = √(2ax² + bx) ≠ 2√ax² + √bx

In the above example, it is crucial to apply the square root to the entire term, including the coefficient 2 and the variable x².

Here is a table summarizing the rules for applying fractional exponents to parentheses:

Expression	Simplified Form
(a + b)^1/n	√(a + b)
(ax² + bx)^1/n	√(ax² + bx)
(2ax² + bx)^1/2	√(2ax² + bx)

Applications of Exponential Expressions

Biology

Exponential functions are used to model population growth, where the rate of growth is proportional to the size of the population. Bacteria, for example, reproduce at a rate proportional to their population size, and thus their growth can be modeled with the function P(t) = P0 * e^(rt), where P0 is the initial population, t represents time, and r is the rate of growth.

Finance

Compound interest accrues through exponential growth, where the interest earned in each period is added to the principal, and then interest is earned on the new total. The formula for compound interest is A = P * (1 + r/n)^(nt), where A is the total amount after n compounding periods, P is the initial principal, r is the annual interest rate, n is the number of compounding periods per year, and t represents the number of years.

Physics

Radioactive decay follows an exponential decay pattern, where the amount of radioactive material decreases at a rate proportional to the amount present. The formula for radioactive decay is A = A0 * e^(-kt), where A0 is the initial amount of radioactive material, A is the amount remaining after time t, and k is the decay constant.

Chemistry

Exponential functions are used in chemical kinetics to model the rate of reactions. The Arrhenius equation, for example, models the rate constant of a reaction as a function of temperature, and the equation for the integrated rate law of a second-order reaction is an exponential decay.

Number 9

The number 9 has several notable applications in mathematics and science.

It is the square of 3 and the cube of 1.
It is the number of planets in our solar system.
It is the atomic number of fluorine.
It is the number of vertices in a regular nonagon.
It is the number of faces on a regular nonahedron.
It is the number of edges on a regular octahedron.
It is the number of faces on a regular truncated octahedron.
It is the number of vertices on a regular truncated dodecahedron.
It is the number of faces on a regular snub cube.
It is the number of vertices on a regular snub dodecahedron.

Property	Value
Square	81
Cube	729
Square root	3
Cube root	1

Common Errors and Pitfalls

1. Mismatching Parentheses

Ensure that every opening parenthesis has a corresponding closing parenthesis, and vice versa. Overlooked or extra parentheses can lead to incorrect results.

2. Incorrect Parenthesis Placement

Pay attention to the placement of parentheses within the power expression. Misplaced parentheses can significantly alter the order of operations and the final result.

3. Confusing Exponents and Parentheses

Distinguish between exponents and parentheses. Exponents are superscripts that denote repeated multiplication, while parentheses group mathematical operations.

4. Order of Operations Errors

Recall the order of operations: parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction. Failure to follow this order can result in incorrect calculations.

10. Complex Expressions with Multiple Parentheses

When dealing with complex expressions containing multiple sets of parentheses, it’s crucial to simplify the expression in a step-by-step manner. Use the order of operations to evaluate the innermost parentheses first, working your way outward until the entire expression is simplified.

To avoid errors when evaluating complex expressions with multiple parentheses, consider the following strategies:

Strategy	Description
Use Parenthesis Notation	Enclose entire expressions within parentheses to clarify the order of operations.
Simplify in Steps	Evaluate the innermost parentheses first and gradually work your way outward.
Use a Calculator	Double-check your calculations using a scientific calculator to ensure accuracy.

How To Solve Parentheses Raised To A Power

When solving parentheses raised to a power, it is important to follow the order of operations. First, solve any parentheses within the parentheses. Then, solve any exponents within the parentheses. Finally, raise the entire expression to the power.

For example, to solve (2 + 3)^2, first solve the parentheses: 2 + 3 = 5. Then, square the result: 5^2 = 25.

Here are some additional examples of solving parentheses raised to a power:

(4 – 1)^3 = 3^3 = 27
(2x + 3)^2 = 4x^2 + 12x + 9
[(x – 2)(x + 3)]^2 = (x^2 + x – 6)^2