Laws Of Exponents
What is the product of powers rule in exponents?
The product of powers property tells that different bases raised to the same power can be grouped in a product that will be raised to that power. It also tells that the product between two powers of the same base will maintain the base and sum the exponents of the powers being multiplied.
How do you calculate the volume of a rectangular prism?
V=whl
What is the Pythagorean Theorem, and how is it represented mathematically?
Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2.
What is the scientific notation, and why is it useful in mathematics and science?
Scientific notation is a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 ✕ 10^8. Created by Sal Khan and CK-12 Foundation.
What is the difference between permutations and combinations?
A permutation is used for the list of data (where the order of the data matters) and the combination is used for a group of data (where the order of data doesn't matter).
How does the power of a power rule work in exponents?
The power of a power rule states that if a base raised to a power is being raised to another power, the exponents are multiplied and the base remains the same.
What is the formula for the volume of a cylinder?
Cylinder's volume is given by the formula, πr2h, where r is the radius of the circular base and h is the height of the cylinder.
How can you use the Pythagorean theorem to find the length of the hypotenuse in a right triangle?
The hypotenuse formula simply takes the Pythagorean theorem and solves for the hypotenuse, c . To solve for the hypotenuse, we simply take the square root of both sides of the equation a² + b² = c² and solve for c . When doing so, we get c = √(a² + b²) .
How do you convert a large number into scientific notation?
To convert any number into scientific notation, you write the non-zero digits, placing a decimal after the first non-zero digit. Then, you count the number of digits you need to move the beginning decimal to get to where your decimal is now. If you move the decimal to the left, then your power is positive.
How do you solve a quadratic equation using the quadratic formula?
The quadratic formula helps us solve any quadratic equation. First, we bring the equation to the form ax²+bx+c=0, where a, b, and c are coefficients. Then, we plug these coefficients in the formula: (-b±√(b²-4ac))/(2a).
Can you explain the quotient of powers rule with an example?
The quotient rule for exponents is that if dividing powers with the same base, the exponents should be subtracted. For example, h^6 divided by h^2 = h^(6-2) = h^4.
Can you explain how to find the volume of a sphere?
The formula for the volume of a sphere is V = 4/3 π r³, where V = volume and r = radius. The radius of a sphere is half its diameter. So, to calculate the surface area of a sphere given the diameter of the sphere, you can first calculate the radius, then the volume.
Can you provide an example of how to use the Pythagorean theorem to solve for a missing side length in a right triangle?
The Pythagorean theorem states that a2 + b2 = c2 in a right triangle where c is the longest side. You can use this equation to figure out the length of one side if you have the lengths of the other two. The figure shows two right triangles that are each missing one side's measure.
Can you explain how to convert a small decimal number into scientific notation?
tep 1: Place a decimal between the 1 and the 3 to create a number between 1 and 10. The coefficient will be . Step 2: The decimal moved 8 places to the left, so the exponent will be . Step 3: When we put these values into scientific notation we get: 1.34 × 10 8.
What are the properties of different types of triangles?
Equilateral Triangle: All the sides are equal and all the three angles equal to 60°. Acute Angled Triangle: A triangle having all its angles less than 90°. Right Angled Triangle: A triangle having one of the three angles exactly 90°. Obtuse Angled Triangle: A triangle having one of the three angles more than 90°.
What is the zero exponent rule, and why does any number raised to the power if zero equal one?
The multiplicative identity is the number 1, because for any other number x, 1*x = x. So, the reason that any number to the zero power is one because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1.
How do you determine the volume of a cone?
The formula for the volume of a cone is ⅓ 𝜋r2h cubic units, where r is the radius of the circular base and h is the height of the cone.
How does the Pythagorean theorem apply to real-world scenarios or problems?
Pythagoras Theorem is used to find the shortest distance in Navigation. The concept of the Pythagoras Theorem is also used in Interior Designing. Pythagoras theorem is used to find the third side of a right-angled triangle when 2 sides are given.
How do you multiply and divide numbers in scientific notation?
Since all number in scientific notation have base 10, we can always multiply them and divide them. To multiply two numbers in scientific notation, multiply their coefficients and add their exponents. To divide two numbers in scientific notation, divide their coefficients and subtract their exponents.
How do you find the are and circumference of a circle?
Use the formulas to calculate the circumference and area: c = 2πr and A = πr² . The circumference should equal c = 2π × 8 cm = 50.265 cm . The area should equal A = π × (8 cm)² = 201.06 cm² . To check your results, input the 8 cm radius in the calculator.
How do you simplify expressions with negative exponents?
To simplify expressions with negative exponents, you can move the base with the negative exponent to the denominator and change the sign of the exponent to positive.
What steps are involved in calculating the volume of a complex shape made up of multiple simple shapes?
Use the appropriate dimensions and area formulas to calculate the area of each simple shape. Then, add the areas together to find the combined surface area of the composite figure. Lastly, to find the volume of a composite shape, multiply the figure's surface area by its height.
What are the conditions under which the Pythagorean is applicable?
It only works for right triangles. To solve the Pythagorean theorem, we need to know the lengths of at least two sides of a right triangle. The Pythagorean theorem formula can be used to find the length of the shorter sides of a right triangle or the longest side, called the hypotenuse.
What are some real-world applications of scientific notation?
Scientific notation is widely used to express very large or very small numbers in a compact and standardized form. Here are some real-world examples: Astronomical distances: The distance from the Earth to the sun is approximately 93 million miles. In scientific notation, this is written as 9.3 x 107 miles
Can you explain the concept of limits in calculus?
The limits are defined as the value that the function approaches as it goes to an x value. Using this definition, it is possible to find the value of the limits given a graph. A few examples are below: In general, you can see that these limits are equal to the value of the function.