Exponential Equations With Powers of X

The Organic Chemistry Tutor
25 Jan 202005:22
EducationalLearning
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TLDRThis educational video script presents a mathematical problem involving exponential equations: finding the value of x when X raised to the power of X cubed equals 729. The presenter guides viewers through the process of solving the problem by cubing both sides of the equation, utilizing exponent rules, and simplifying the expression to a more manageable form. The solution involves recognizing that 729 is 9^3, and thus the equation simplifies to x^3 raised to the x^3 power, equating to 9^9. The key insight is that x^3 must equal 9, leading to the conclusion that x is the cube root of 9. The script concludes with a verification step, encouraging viewers to plug the answer back into the original equation to confirm its correctness.

Takeaways
  • 🧐 The math problem presented is \( X^X^3 = 729 \), where \( X \) is the unknown variable.
  • 📚 The first step is to cube both sides of the equation to simplify the problem.
  • 🔢 Reviewing exponent rules, it's explained that \( a^2 \) raised to the third power equals \( a^6 \) because \( 2 \times 3 = 6 \).
  • 🔄 The script demonstrates how to manipulate exponents, showing that \( X^3 \) raised to the third power can be written as \( X^{3 \times 3} \) or \( X^9 \).
  • 📉 The expression is then rewritten as \( X^9 \) raised to the power of \( X^3 \), using the rule of exponents.
  • 📈 Recognizing that 729 is \( 9^3 \), the problem is simplified to \( X^9 \) raised to \( X^3 \) equals \( 9^9 \).
  • 💡 The insight is that when an exponent is raised to another exponent, the bases can be multiplied, leading to \( X^9 \) equals \( 9^9 \).
  • 📌 It's concluded that \( X^3 \) must be equal to 9, as both sides of the equation are now the same base raised to the same power.
  • 👉 To find \( X \), both sides of the equation \( X^3 = 9 \) are raised to the power of \( \frac{1}{3} \), which is the cube root.
  • 📝 The final answer is \( X = \sqrt[3]{9} \), which is the cube root of 9.
  • 🔄 The solution is verified by plugging the value of \( X \) back into the original expression to confirm it equals 729.
Q & A

  • What is the original math problem presented in the transcript?

    -The original math problem is to find the value of x in the equation X raised to the power of X cubed equals 729.

  • Why do we need to raise both sides of the equation to the third power?

    -Raising both sides to the third power is done to simplify the equation by reducing the exponents and making the problem easier to solve.

  • Can you explain the concept of raising an exponent to another exponent?

    -When you raise an exponent to another exponent, you multiply the exponents together. For example, a^(m^n) is equivalent to (a^m)^n.

  • What is the significance of the number 729 in the context of this problem?

    -The number 729 is significant because it is a perfect cube (9^3), which allows us to simplify the equation by expressing it as 9 raised to the third power.

  • How does the script suggest rewriting the expression X^(X^3)^3?

    -The script suggests rewriting the expression as X^(3*X^3) or as (X^3)^X, which is equivalent to 3 times X cubed.

  • What mathematical property is used to rewrite the expression X^(3*X^3) as (X^3)^X?

    -The property of exponents that allows us to rewrite X^(3*X^3) as (X^3)^X is that when you raise one exponent to another, you can multiply the exponents.

  • How does the script determine that X^3 equals 9?

    -The script determines that X^3 equals 9 by recognizing that 729 is 9 cubed, and thus, by setting X^3 equal to 9, we can simplify the equation.

  • What is the next step after determining that X^3 equals 9?

    -The next step is to find the value of x by raising both sides of the equation X^3 = 9 to the power of 1/3, which gives us x = 9^(1/3).

  • What is the final answer to the original math problem?

    -The final answer to the original math problem is that x equals the cube root of 9.

  • How can we verify the solution to the original math problem?

    -We can verify the solution by plugging the value of x (the cube root of 9) back into the original expression X^(X^3) and checking if it equals 729.

Outlines
00:00
🧐 Solving the Exponential Equation X^(X^3) = 729

The video begins by presenting a challenging math problem: finding the value of x in the equation X^(X^3) = 729. The host encourages viewers to attempt the problem and hints at raising both sides to the third power as a strategy. The explanation continues with a review of exponent rules, demonstrating how to multiply exponents and reverse the order of operations. The key insight is to recognize that 729 is 9 cubed, which simplifies the equation to X^(X^3) = 9^9. By equating the bases of the exponents, it's deduced that X^3 must be 9. The final step involves taking the cube root of both sides to solve for x, revealing that x is the cube root of 9. The solution is verified by plugging the value back into the original equation.

05:01
📚 Verification of the Solution for the Exponential Equation

This paragraph focuses on verifying the solution to the exponential equation. The host suggests that viewers can use a calculator to confirm the solution by raising the cube root of 9 to the power of itself cubed, which should equal 729. The paragraph concludes with a brief acknowledgment of the successful completion of the problem-solving process and thanks the viewers for watching. This step ensures that viewers understand the verification process and can independently confirm the accuracy of the solution.

Mindmap
Keywords
💡Exponents
Exponents are mathematical symbols that indicate the number of times a base number is multiplied by itself. In the context of the video, exponents are central to solving the problem as the script explains how to manipulate equations involving exponents to simplify and solve for the variable x. For example, the script mentions 'a squared and we raise it to the third power, this is equal to a to the sixth power' to demonstrate the rules of exponents.
💡Cube
A cube, in mathematics, refers to raising a number to the power of three (i.e., multiplying the number by itself three times). The video script uses the term 'cube' to describe the operation 'X raised to the X cubed', which is a key part of the equation that needs to be solved. The script also explains that '729 is a cube of what number', using the cube concept to simplify the equation.
💡Equation
An equation is a mathematical statement that asserts the equality of two expressions. In the video, the equation 'X raised to the X cubed is equal to 729' is the central problem that needs to be solved. The script guides viewers through the process of manipulating this equation to find the value of x.
💡Raising to a Power
Raising to a power is the operation of multiplying a number by itself a certain number of times, as indicated by the exponent. The video script uses this concept to transform the original equation into a more solvable form by 'raising both sides to the third power' and later by rewriting the equation as 'x to the Third raised to the X cubed'.
💡Cube Root
A cube root of a number is a value that, when multiplied by itself three times, gives the original number. The script introduces the concept of the cube root when solving for x, stating 'to get X by itself what we need to do is, raise both sides to the 1 3 power', which simplifies to finding the cube root of 9 to determine the value of x.
💡Multiplication of Exponents
The multiplication of exponents is a rule in mathematics that states when you multiply powers with the same base, you can add the exponents. The video script uses this rule when explaining 'X cubed times 3 or 3x cubed', which simplifies to 'X raised to the 3 times x cubed'.
💡Fractional Exponent
A fractional exponent represents a root of a number. In the video, the script converts the exponent '1/3' into a radical, stating '9 raised to the one third we can convert the fractional exponent into a radical, so this is the cube root of nine', which is used to solve for x.
💡Base
In an exponential expression, the base is the number that is being raised to a power. The script refers to 'a' as the base in the example 'if we have a squared and we raise it to the third power', and 'X' is the base in the main problem 'X raised to the X cubed'.
💡Simplification
Simplification in mathematics involves making a complex expression or equation easier to understand or solve. The video script demonstrates simplification by transforming the original problem into a more manageable form, such as changing 'X cubed raised to the X cubed' to '9 raised to the ninth power'.
💡Root
A root of a number is a value that, when raised to a certain power, gives the original number. The script refers to finding the cube root of 9 to solve for x, indicating that 'the cube root of nine' is the solution to the equation.
Highlights

The math problem presented is X raised to the power of X cubed equals 729.

Suggestion to take a minute to try solving the problem before moving forward.

Invitation to check out more challenging math problems in the description section.

Introduction to the first step of solving the problem by raising both sides to the third power.

Explanation of the reasoning behind cubing both sides of the equation.

Review of exponent rules with examples of squaring and cubing.

Clarification on how to handle the expression X cubed raised to the third power.

Demonstration of rewriting the expression to simplify the problem.

Identification of 729 as 9 cubed, simplifying the equation.

Explanation of multiplying exponents when raising one to another.

Transformation of the equation to have a number raised to the same number.

Realization that X cubed equals 9 from the simplified equation.

Process of isolating X by raising both sides to the power of 1/3.

Conversion of the fractional exponent into a radical to find X.

Final answer that X is the cube root of 9.

Validation of the solution by plugging it back into the original expression.

Conclusion and thanks for watching, summarizing how to solve exponential equations.

Transcripts

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