What Is The Value Of $-5^6$?A. { -15,625$}$B. { -30$}$C. 30D. 15,625
Understanding the Concept of Exponents
When dealing with exponents, it's essential to understand the rules and properties that govern them. In this case, we're tasked with finding the value of $-5^6$. To approach this problem, we need to recall the rule that states when a negative number is raised to a power, the negative sign is moved to the exponent, and the exponent is then applied to the absolute value of the number.
Applying the Rule for Negative Exponents
According to the rule, $-5^6$ can be rewritten as $(-5)^6$. This is because the negative sign is moved to the exponent, and the exponent is then applied to the absolute value of the number, which is 5.
Evaluating the Expression
Now that we have rewritten the expression as $(-5)^6$, we can evaluate it by applying the exponent to the absolute value of the number. This means we need to calculate $5^6$ first and then apply the negative sign.
Calculating $5^6$
To calculate $5^6$, we need to multiply 5 by itself 6 times. This can be done using the formula for exponentiation:
5^6 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 </span></p> <h2>Performing the Multiplication</h2> <p>Now that we have the formula, we can perform the multiplication:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>5</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>25</mn></mrow><annotation encoding="application/x-tex">5 \times 5 = 25 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">25</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>25</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>125</mn></mrow><annotation encoding="application/x-tex">25 \times 5 = 125 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">25</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">125</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>125</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>625</mn></mrow><annotation encoding="application/x-tex">125 \times 5 = 625 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">125</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">625</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>625</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>3125</mn></mrow><annotation encoding="application/x-tex">625 \times 5 = 3125 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">625</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3125</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>3125</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>15625</mn></mrow><annotation encoding="application/x-tex">3125 \times 5 = 15625 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3125</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">15625</span></span></span></span></span></p> <h2>Applying the Negative Sign</h2> <p>Now that we have calculated $5^6$, we can apply the negative sign to get the final result:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>5</mn><msup><mo stretchy="false">)</mo><mn>6</mn></msup><mo>=</mo><mo>−</mo><mn>15625</mn></mrow><annotation encoding="application/x-tex">(-5)^6 = -15625 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">5</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">15625</span></span></span></span></span></p> <h2>Conclusion</h2> <p>In conclusion, the value of $-5^6$ is $-15625$. This is because we applied the rule for negative exponents, calculated $5^6$, and then applied the negative sign to get the final result.</p> <h2>Answer Options</h2> <p>Based on our calculation, we can see that the correct answer is:</p> <p>A. <span class="katex-error" title="ParseError: KaTeX parse error: Expected '}', got 'EOF' at end of input: {" style="color:#cc0000">{</span>-15,625$}{{content}}lt;/p> <p>This is the only option that matches our final result.</p> <h2>Discussion</h2> <p>This problem requires a good understanding of exponents and how to apply the rules for negative exponents. It's essential to remember that when a negative number is raised to a power, the negative sign is moved to the exponent, and the exponent is then applied to the absolute value of the number. This can be a challenging concept to grasp, but with practice and patience, it becomes easier to understand and apply.</p> <h2>Tips and Tricks</h2> <p>Here are some tips and tricks to help you solve problems like this:</p> <ul> <li>Make sure to understand the rules for exponents, including how to apply the negative sign.</li> <li>Use the formula for exponentiation to calculate the value of the expression.</li> <li>Perform the multiplication carefully to avoid errors.</li> <li>Apply the negative sign to the final result to get the correct answer.</li> </ul> <p>By following these tips and tricks, you can improve your skills and become more confident in solving problems like this.<br/></p> <h2>Frequently Asked Questions</h2> <p>In this article, we'll address some common questions and concerns related to exponents and negative numbers. Whether you're a student, teacher, or simply looking to improve your math skills, this Q&A section is designed to provide you with the information and guidance you need.</p> <h2>Q: What is the rule for negative exponents?</h2> <p>A: The rule for negative exponents states that when a negative number is raised to a power, the negative sign is moved to the exponent, and the exponent is then applied to the absolute value of the number. For example, $-5^6$ can be rewritten as $(-5)^6$.</p> <h2>Q: How do I calculate $a^b$?</h2> <p>A: To calculate $a^b$, you need to multiply $a$ by itself $b$ times. For example, to calculate $5^6$, you would multiply 5 by itself 6 times: $5 \times 5 \times 5 \times 5 \times 5 \times 5$.</p> <h2>Q: What is the difference between $a^b$ and $(a^b)$?</h2> <p>A: $a^b$ represents the exponentiation of $a$ to the power of $b$, while $(a^b)$ represents the exponentiation of $a$ to the power of $b$, with the result then being raised to the power of $b$. For example, $5^6$ is different from $(5^6)$.</p> <h2>Q: Can I simplify expressions with negative exponents?</h2> <p>A: Yes, you can simplify expressions with negative exponents by applying the rule for negative exponents. For example, $-5^6$ can be rewritten as $(-5)^6$, which can then be simplified to $-15625$.</p> <h2>Q: How do I handle negative numbers with exponents?</h2> <p>A: When dealing with negative numbers and exponents, it's essential to remember that the negative sign is moved to the exponent, and the exponent is then applied to the absolute value of the number. For example, $-5^6$ can be rewritten as $(-5)^6$.</p> <h2>Q: Can I use a calculator to calculate exponents?</h2> <p>A: Yes, you can use a calculator to calculate exponents. However, it's essential to understand the rules and properties of exponents to ensure that you're using the calculator correctly.</p> <h2>Q: What are some common mistakes to avoid when working with exponents?</h2> <p>A: Some common mistakes to avoid when working with exponents include:</p> <ul> <li>Forgetting to move the negative sign to the exponent</li> <li>Not applying the exponent to the absolute value of the number</li> <li>Confusing $a^b$ with $(a^b)$</li> <li>Not simplifying expressions with negative exponents</li> </ul> <h2>Q: How can I practice and improve my skills with exponents?</h2> <p>A: To practice and improve your skills with exponents, try the following:</p> <ul> <li>Work through examples and exercises in your textbook or online resources</li> <li>Practice calculating exponents with different bases and exponents</li> <li>Use online calculators or software to check your work and identify areas for improvement</li> <li>Seek help from a teacher or tutor if you're struggling with a particular concept or problem</li> </ul> <p>By following these tips and practicing regularly, you can improve your skills and become more confident in working with exponents and negative numbers.</p>