Evaluate. $ {3}^{2} $ { - 4}^{ - 2} $ \frac{1}{2} {?}^{ - 5} $
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The given mathematical expression is a combination of exponentiation, negative exponents, and fractions. To evaluate this expression, we need to follow the order of operations (PEMDAS) and apply the rules of exponents.
Understanding the Components of the Expression
The given expression consists of three main components:
- Exponentiation: The first two components are exponentiated expressions, where the base is raised to a power. In the first component, 3 is raised to the power of 2, and in the second component, -4 is raised to the power of -2.
- Negative Exponents: The second component involves a negative exponent, which can be rewritten as a positive exponent by taking the reciprocal of the base.
- Fraction: The third component is a fraction, where 1 is divided by 2.
Evaluating the Exponentiated Components
Let's start by evaluating the exponentiated components:
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3^2: This is a simple exponentiation, where 3 is raised to the power of 2. Using the rule of exponents, we can calculate this as follows:
- 3^2 = 3 × 3 = 9
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(-4)^(-2): This involves a negative exponent, which can be rewritten as a positive exponent by taking the reciprocal of the base. The reciprocal of -4 is -1/4, and raising it to the power of 2 gives us:
- (-4)^(-2) = (-1/4)^2 = 1/16
Evaluating the Fraction
The third component is a fraction, where 1 is divided by 2. This can be evaluated as follows:
- 1/2 = 0.5
Evaluating the Entire Expression
Now that we have evaluated the individual components, let's put them together to evaluate the entire expression. The expression is:
- 3^2 × (-4)^(-2) × 1/2 × ?^(-5)
However, there is a missing component in the expression, denoted by ?^(-5). To evaluate the entire expression, we need to know the value of this component.
Conclusion
In conclusion, the given mathematical expression involves exponentiation, negative exponents, and fractions. To evaluate this expression, we need to follow the order of operations (PEMDAS) and apply the rules of exponents. However, the expression is incomplete, and we need to know the value of the missing component to evaluate the entire expression.
Frequently Asked Questions
Q: What is the order of operations (PEMDAS)?
A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I handle negative exponents?
A: Negative exponents can be rewritten as positive exponents by taking the reciprocal of the base. For example, (-4)^(-2) can be rewritten as (-1/4)^2.
Q: What is the rule of exponents?
A: The rule of exponents states that when multiplying two numbers with the same base, we can add their exponents. For example, 2^3 × 2^4 = 2^(3+4) = 2^7.
Final Answer
The final answer to the given mathematical expression is incomplete, as the value of the missing component ?^(-5) is not provided. To evaluate the entire expression, we need to know the value of this component.
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In the previous article, we evaluated a mathematical expression involving exponentiation, negative exponents, and fractions. However, the expression was incomplete, and we needed to know the value of the missing component to evaluate the entire expression. In this article, we will answer some frequently asked questions (FAQs) related to mathematical expressions.
Q: What is the order of operations (PEMDAS)?
A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I handle negative exponents?
A: Negative exponents can be rewritten as positive exponents by taking the reciprocal of the base. For example, (-4)^(-2) can be rewritten as (-1/4)^2.
Q: What is the rule of exponents?
A: The rule of exponents states that when multiplying two numbers with the same base, we can add their exponents. For example, 2^3 × 2^4 = 2^(3+4) = 2^7.
Q: How do I evaluate expressions with multiple operations?
A: When evaluating expressions with multiple operations, we need to follow the order of operations (PEMDAS). We start by evaluating expressions inside parentheses, then evaluate any exponential expressions, followed by multiplication and division operations, and finally addition and subtraction operations.
Q: What is the difference between a fraction and a decimal?
A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. For example, 1/2 is a fraction. A decimal is a way of expressing a fraction as a number with a point (.) separating the whole number part from the fractional part. For example, 0.5 is a decimal.
Q: How do I convert a fraction to a decimal?
A: To convert a fraction to a decimal, we can divide the numerator by the denominator. For example, to convert 1/2 to a decimal, we can divide 1 by 2, which gives us 0.5.
Q: What is the difference between a positive exponent and a negative exponent?
A: A positive exponent is an exponent that is greater than or equal to 1. For example, 2^3 is a positive exponent. A negative exponent is an exponent that is less than 1. For example, 2^(-3) is a negative exponent.
Q: How do I handle negative exponents in fractions?
A: When a negative exponent is present in a fraction, we can rewrite the fraction as a positive exponent by taking the reciprocal of the base. For example, (1/2)^(-3) can be rewritten as (2/1)^3.
Q: What is the rule for multiplying fractions?
A: When multiplying fractions, we can multiply the numerators together and multiply the denominators together. For example, (1/2) × (3/4) = (1 × 3) / (2 × 4) = 3/8.
Q: What is the rule for dividing fractions?
A: When dividing fractions, we can invert the second fraction and multiply. For example, (1/2) ÷ (3/4) = (1/2) × (4/3) = (1 × 4) / (2 × 3) = 4/6 = 2/3.
Q: How do I simplify a fraction?
A: To simplify a fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD). For example, to simplify 6/8, we can divide both 6 and 8 by 2, which gives us 3/4.
Q: What is the difference between a rational number and an irrational number?
A: A rational number is a number that can be expressed as a ratio of two integers. For example, 3/4 is a rational number. An irrational number is a number that cannot be expressed as a ratio of two integers. For example, the square root of 2 is an irrational number.
Q: How do I determine if a number is rational or irrational?
A: To determine if a number is rational or irrational, we can try to express it as a ratio of two integers. If we can express it as a ratio of two integers, then it is a rational number. If we cannot express it as a ratio of two integers, then it is an irrational number.
Q: What is the difference between a real number and an imaginary number?
A: A real number is a number that can be expressed on the number line. For example, 3 is a real number. An imaginary number is a number that cannot be expressed on the number line. For example, the square root of -1 is an imaginary number.
Q: How do I determine if a number is real or imaginary?
A: To determine if a number is real or imaginary, we can try to express it on the number line. If we can express it on the number line, then it is a real number. If we cannot express it on the number line, then it is an imaginary number.
Q: What is the difference between a complex number and a real number?
A: A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. For example, 3 + 4i is a complex number. A real number is a number that can be expressed on the number line. For example, 3 is a real number.
Q: How do I determine if a number is complex or real?
A: To determine if a number is complex or real, we can try to express it in the form a + bi. If we can express it in this form, then it is a complex number. If we cannot express it in this form, then it is a real number.
Q: What is the difference between a polynomial and a rational function?
A: A polynomial is an expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. For example, x^2 + 3x - 4 is a polynomial. A rational function is an expression that consists of a polynomial divided by another polynomial. For example, (x^2 + 3x - 4) / (x + 1) is a rational function.
Q: How do I determine if an expression is a polynomial or a rational function?
A: To determine if an expression is a polynomial or a rational function, we can try to express it as a polynomial divided by another polynomial. If we can express it in this form, then it is a rational function. If we cannot express it in this form, then it is a polynomial.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation that can be expressed in the form ax + b = c, where a, b, and c are constants. For example, 2x + 3 = 5 is a linear equation. A quadratic equation is an equation that can be expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants. For example, x^2 + 4x + 4 = 0 is a quadratic equation.
Q: How do I determine if an equation is linear or quadratic?
A: To determine if an equation is linear or quadratic, we can try to express it in the form ax + b = c or ax^2 + bx + c = 0. If we can express it in the form ax + b = c, then it is a linear equation. If we can express it in the form ax^2 + bx + c = 0, then it is a quadratic equation.
Q: What is the difference between a system of linear equations and a system of quadratic equations?
A: A system of linear equations is a set of equations that can be expressed in the form ax + b = c, where a, b, and c are constants. For example, 2x + 3 = 5 and x - 2 = 3 is a system of linear equations. A system of quadratic equations is a set of equations that can be expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants. For example, x^2 + 4x + 4 = 0 and x^2 - 2x - 3 = 0 is a system of quadratic equations.
Q: How do I determine if a system of equations is linear or quadratic?
A: To determine if a system of equations is linear or quadratic, we can try to express each equation in the form ax + b = c or ax^2 + bx + c = 0. If all the equations can be expressed in the form ax + b = c, then the system is a system of linear equations. If any of the equations can be expressed in the form ax^2 + bx + c = 0, then the system is a