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Essential Concepts

• Decimals are a way to express parts of a whole or fractions, using a decimal point. They help us represent values between whole numbers and are important for accurate measurements and calculations.
• To name a decimal number, we read the number to the left of the decimal point, say “and” for the decimal point, read the number to the right as a whole number, and name the decimal place of the last digit.
• To write a decimal number from its name, we look for the word “and” to locate the decimal point, mark the number of decimal places, translate the words before “and” into the whole number, write a “[latex]0[/latex]” if there is no “and,” translate the words after “and” into the number after the decimal point, and fill in zeros as needed.
• To convert a decimal number to a fraction or mixed number, look at the number to the left of the decimal. If it’s zero, the decimal becomes a proper fraction; if it’s not zero, it becomes a mixed number. Write the whole number part, determine the place value of the final digit, and write the fraction with the numerator as the numbers to the right of the decimal and the denominator as the place value of the final digit. Simplify the fraction if possible.
• To round decimals, find the given place value and underline the digit to the right. If the digit is greater than or equal to [latex]5[/latex], add [latex]1[/latex] to the digit in the given place value; if not, leave it as it is. Rewrite the number, removing all digits to the right of the given place value.
• To order decimals, first make sure they have the same number of decimal places. If not, add zeros at the end of the one with fewer digits to match them. Then compare the numbers to the right of the decimal point as if they were whole numbers, and use the appropriate inequality sign to determine the order.
• To determine if a number is a solution to an equation, replace the variable in the equation with the number. Then simplify both sides of the equation. If the resulting equation is true, the number is a solution. If it’s not true, the number is not a solution.
• There are four main properties of equality:
  • Addition Property of Equality
    • For any numbers [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex]. If [latex]a=b[/latex], then [latex]a+c=b+c[/latex].
  • Subtraction Property of Equality
    • For any numbers [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex]. If [latex]a=b[/latex], then [latex]a-c=b-c[/latex].
  • The Multiplication Property of Equality
    • For any numbers [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex]. If [latex]a=b[/latex], then [latex]ac=bc[/latex].
  • The Division Property of Equality
    • For any numbers [latex]a,b,\text{and }c,\text{and }c\ne 0[/latex]. If [latex]a=b[/latex], then [latex]{\Large\frac{a}{c}}={\Large\frac{b}{c}}[/latex].
• A percent is a way of comparing a part to a whole. It is calculated by dividing the part by the whole and multiplying by [latex]100[/latex]. To work with percentages, we can write them as decimals or fractions.
• A rate compares two quantities and is expressed as a fraction. A unit rate is a special type of rate where the denominator is one. A proportion equation is an equation that shows that two rates or ratios are equal to each other.
• Absolute change compares the difference between two quantities, ignoring the direction of change. It is calculated by subtracting the starting quantity from the ending quantity, and the result has the same units as the original quantity. Relative change, on the other hand, measures the percent change between two quantities. It is determined by dividing the absolute change by the starting quantity and expressing it as a percentage. The starting quantity is referred to as the base of the percent change.
• Exponential notation is a way to write numbers using a base and an exponent. The base tells us the number being multiplied, and the exponent represents how many times the base is multiplied by itself.
• Negative exponents indicate that the number is in the denominator of a fraction and can be written as the reciprocal of the positive exponent.
• The product rule for exponents: [latex]\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}[/latex].
• The quotient rule for exponents: [latex]\displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[/latex].
• The power rule for exponents: [latex]\displaystyle{\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}[/latex].
• The product raised to a power rule: [latex]\left(ab\right)^{x}=a^{x}\cdot{b^{x}}[/latex]
• The quotient raised to a power rule: [latex]\displaystyle {\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}[/latex].
• Rules for [latex]0[/latex] and [latex]1[/latex]: [latex]n^{1}=n[/latex] and [latex]n^{0}=1[/latex].
• The negative rule of exponents: [latex]a^{-m}=\frac{1}{a^{m}}[/latex] and [latex]\frac{1}{a^{-m}}=a^{m}[/latex].
• Scientific notation is a way to write very large or very small numbers in a shorter form. It uses a coefficient (a number between [latex]1[/latex] and [latex]10[/latex]) multiplied by [latex]10[/latex] raised to a power (an integer). Positive powers are used for large numbers, while negative powers are used for small numbers.
• When multiplying numbers in scientific notation, we multiply the coefficients together and add the powers of [latex]10[/latex]. For division, we divide the coefficients and subtract the powers of [latex]10[/latex].
• To evaluate an algebraic expression, you substitute the given values for the variables in the expression. Then you simplify the expression using the order of operations, just like you would with regular math problems. If there is more than one variable, you replace each variable with its assigned value and simplify the expression in the same way.
• To solve a multi-step equation, follow these steps:
  • (Optional) If there are fractions or decimals, you can multiply the whole equation by a number to get rid of them.
  • Simplify both sides of the equation by removing parentheses and combining similar terms.
  • Add or subtract terms to isolate the variable term. You might need to move terms around to achieve this.
  • Multiply or divide to get the variable by itself.
  • Finally, check if the solution you found makes the equation true.
• An equation can have different outcomes based on the values it contains. Here are the possibilities:
  • One Solution: Sometimes an equation has only one value that makes it true. This means there is only one answer or solution that satisfies the equation.
  • No Solution: In some cases, there are no values that make the equation true. This means the equation has no solution, and there is no answer that satisfies the given conditions.
  • Many Solutions: Occasionally, an equation can have multiple values that make it true. This means there are many possible solutions or answers that satisfy the equation.

Glossary

algebraic expression

a mathematical phrase or combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division

constant

a fixed value or a number that does not change in a particular context

decimals

represent fractions or parts of a whole, based on powers of ten, using a point known as a decimal point

equation

a mathematical statement that shows the equality of two expressions, typically separated by an equal sign

equivalent decimals

two decimals that convert to equivalent fractions

formula

a mathematical expression that represents a relationship or a rule between variables or quantities

proportion equation

an equation showing the equivalence of two rates or ratios

rate

the ratio (fraction) of two quantities

ratio

a comparison of two numbers by division

scientific notation

a way of writing numbers that are very large or very small in a compact and convenient form

unit rate

a rate with a denominator of one

variable

a symbol that represents a value or quantity that can change or vary in a given situation or context

Key Equations

absolute change

[latex]\displaystyle|\text{ending quantity}-\text{starting quantity}|[/latex]

negative rule of exponents

[latex]a^{-m}=\frac{1}{a^{m}}[/latex] and [latex]\frac{1}{a^{-m}}=a^{m}[/latex]

power rule for exponents

[latex]\displaystyle{\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}[/latex].

product raised to a power rule

[latex]\left(ab\right)^{x}=a^{x}\cdot{b^{x}}[/latex]

product rule for exponents

[latex]\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}[/latex].

quotient raised to a power rule

[latex]\displaystyle {\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}[/latex].

quotient rule for exponents

[latex]\displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[/latex].

relative change

[latex]\displaystyle\frac{\text{absolute change}}{\text{starting quantity}}[/latex]